Determining Total System Efficiency

AI Use Disclaimer

• I have used AI throughout the thought process here
• Usage went something like this:
  1. I want to understand the full calculation to determine COP in an HVAC system
  2. Walk through that response over the course of a week (create this page) until I fully understand where the calculations come from and what they mean
     • During this process primarily I am finding online references to corroborate the AI calculations
     • I am also digging deeper into constants
  3. Follow up with a few clarifying questions where I didn’t understand some concepts and couldn’t readily find a source online
  4. After this page was complete I fed it back into AI, it identified:
     • I had a few places where I had mis-typed a number (134 vs 143)
     • An instance where I put a - instead of +
     • I had incorrectly correlated Specific Heat with Specific Gas Constant
• Takeaways:
  • This was a good starting point, AI definitely leaned into industry standard equations, which was fine, but I wanted to dig a level deeper to determine where assumptions / constants come from
  • Equations often didn’t list clearly units so it took a little extra work to understand what was really happening
  • This was very useful as a well consolidated guide for the full-stack calculations, it helped me learn what’s going on quickly
  • Hopefully this consolidated set of calculations is useful for others to have
  • …it could definitely be better organized!

Equation Fundamentals

Throughout these calculations I will be attempting as best as I can to get to two points, the fundamental scientific constants (specific heat of dry air on earth, molar mass ratio of air to water) and measured points (pressure or temperature). I will avoid leaving constants “unexplained” and “raw” as these are areas where potential measurement and tuning based on real measurements could impact those constants.

All that being said, I could also introduce errors to known “solid” constants through bad measurement, so I need to keep this in mind.

Overview

I wanted to understand all the calculations required to determine COP for an HVAC system. I tried in doing this to eliminate any unsubscribed constants and give clarification to the meanings behind terms or their implications where I thought it was not obvious.

Hopefully this is useful to others and creates a good consolidated source of these calculations and how they interact.

…One major caveat is that there are many variations on these equations in terms of their units. I targeted using the most common units in HVAC and/or the units the sensors I have will most likely be read in. Please take note of these units…sometimes they are not “clean” and I might move between some based on the most easily accessible equations online.

Coefficient of Performance (COP)

$COP=\frac{Q_{removed}}{W_{input}}$

$Q_{removed}=Heatremoved,forCooling$

$W_{input}=Wattsused$

COP calculates the efficiency of “Energy In, Energy Out”, HVAC using the compression cycle and mechanical cooling/heating should have a COP of greater than 1. It is using energy to MOVE heat from indoors to outdoors (cooling), so this represents the ratio of Electric Watts used to run the system to how much Heat Watts is removed.

Air-side Enthalpy

How much heat is the air-side removing form the system?

Calculate CFM

Calculate CFM to determine volumetric airflow of the system.

$CFM=K\times \sqrt{\Delta P}$

$K$ can come from coil manufacturer datasheets…where applicable they will provide tables or equations describing the flow rate to pressure drop ratio. This comes with a wet and dry calculation, as the resistance will increase when wet (having run for some time). This must be accounted for in the calculations.

$K$ can also be determined empirically through the use of a calibrated flow meter during operation, although this can be challenging.

Is the coil wet?

If the coil is below the dew point temperature it is assumed that it might contain moisture, Dew-point calculation:

$T_{dew}=\frac{243.04\times \gamma }{17.625-\gamma }$

$Where\gamma =ln(\frac{RH\%}{100})+\frac{17.625\times T_C}{243.04+T_C}$

$T_C$ is the dry-bulb temperature in Celsius

$T_{dew}$ is the dew-point temperature in Celsius

If $T_{coil}<T_{dew}$ water will begin condensing on the coil, although this does not happen instantly and will produce more moisture if the delta is greater.

Mangus Coefficients

I wish I knew better the derivation of the Magnus Coefficents (a=17.625, b=243.04) came from. There are papers calculating these, but it seems a little too complex to be worth digging in deeper for right now.

Alternatively,

Moist Air Enthalpy

Enthalpy of Moist Air Determine enthalpy of the moist air by knowing the humidity of the air (latent heat) and it’s sensible heat (temperature you can feel). Enthalpy per pound of air is:

$h_{moist}=(C_{pa}\times T_{db})+W\times (h_{we}+C_{pw}\times T_{db}$)

$T_{db}$ is the dry bulb temperature (°F)

$h_{dry}$ is the dry air enthalpy ($BTU/lb$)

$W$ is the specific humidity ratio ($\frac{lbwater}{lbdry-air}$)

$C_{pa}$ is the specific heat of air, or $0.240\frac{Btu}{lb\cdot DegF}$

$C_{pw}$ is the specific heat of water, or $0.444\frac{Btu}{lb\cdot DegF}$

$h_{we}$ is the latent heat of vaporization for water, I fit a curve to the below table:

$h_{we}=-5.6992\times T_{db}+1081.8927$ (ignoring pressure, $\frac{Btu}{lb}$)

$T_{db}$	$P_{vap}$	$h_{vap}$
°F	psi	Btu/lbm
32.2	0.0891	1075.2
40	0.1219	1070.7
50	0.1783	1065
60	0.2564	1059.4
70	0.3632	1053.7
80	0.5073	1048
90	0.699	1042.4
100	0.9506	1036.7
110	1.277	1030.9
120	1.695	1025.2
130	2.226	1019.4
140	2.893	1013.6
150	3.723	1007.7
160	4.747	1001.8
170	6	995.87
180	7.519	989.85
190	9.35	983.76
200	11.54	977.6
210	14.14	971.35
212	14.71	970.08
220	17.2	965.02

Specific Humidity Ratio

Specific Humidity Ratio or the ratio of water vapor to dry air

$W={\varphi }_{aw-mm}\times \frac{P_{vapor}}{P_{dry-air}}$

${\varphi }_{aw-mm}$ is the air-to-water molar mass ($\frac{g}{mol}$) ratio or, 0.622

$P_{dry-air}=P_{air}-P_{vapor}$

$\varphi =\frac{P_{v}apor}{P_{sat}}$

$P_{vapor}=\varphi \times P_{sat}$ (Partial Pressure of water if it occupied a vessel by itself)

$\varphi$ is relative humidity 0-1

$P_{sat}$ is the saturation pressure of water, using the Antoine Equation:

$lo{g}_{10}\times p=A-\frac{B}{C+T}$

$P_{sat-mmHg}=10^{(8.07131-\frac{1730.63}{233.426+T_{Celsius}})}mmHg$

$P_{psi}=P_{mmHg}\times 0.01933678$

$P_{air}$ is the static air pressure, this might be $P_{atm}$ but a properly operating HVAC system will maintain some static pressure, $P_{atm}$ should be measured or adjusted for the locations elevation above sea-level.

Relative to ATM, if $P_{air-supply}-P_{air-return}$ is too high it might indicate you have a clogged filter, undersized ducting, oversized fan motor, or excessive blocked/restricted returns, if it is too low, might indicate no filter at all, oversized ducting or undersized fan motor (or not running in reality).

With this calculation you are using the ratio of partial air vapor pressure to determine the dry air vapor pressure (removing it from the atmospheric pressure). Then using that ratio in combination with the molar mass ratio to determine the ratio of the mass/weight of water to dry-air in the air column. Measuring the absolute pressure of the air column at the given point of the humidity will make this calculation more accurate.

Altogether (all Pressure in PSI):

$P_{sat}=10^{(8.07131-\frac{1730.63}{233.426+T_{Celsius}})}\times 0.01933678$

$W=0.622\times \frac{\varphi \times P_{sat}}{P_{air}-(\varphi \times P_{sat})}$

$h_{moist}=(0.24\times T_{db})+W\times ((-5.6992\times T_{db}+1081.8927)+0.444\times T_{db}$)

$h_{moist}$ now describes you the total (latent + sensible) heat contained in the air column in BTU per lb of air.

Total Capacity

Capacity is described (units) as $BTU/hr$ in these equations.

$Q_{total}={\rho }_{air}\times CFM\times 60min/hr\times (h_{return}-h_{supply})$

Calculated air density:

${\rho }_{air}$ is the density of the air, ${\rho }_{air}$ = ${\rho }_{dry-air}$+${\rho }_{vapor}$

$PV=nRT$

$\rho =\frac{n}{V}=\frac{P}{RT}$ (moles per volume = density)

${\rho }_{dry}=\frac{P_{dry-air}}{R_{air}\times T_{rankine}}$, ${\rho }_{vapor}=\frac{P_{vapor}}{R_{vapor}\times T_{rankine}}$

Specific Gas Constant Calculations

$R_{universal}$ is the universal gas constant, $mmo{l}_{air/vapor}$ is the molar mass of vapor and air, these are respectively 10.73 $\frac{psi\cdot f{t}^3}{lbmol\cdot R}$, 28.97 $\frac{lb}{lbmol}$, and 18.015 $\frac{lb}{lbmol}$

$R_{air}=\frac{R_{universal}}{mmo{l}_{air}}=\frac{10.73}{28.97}$ = 0.3704$\frac{psia\cdot f{t}^3}{lb\cdot R}$

$R_{vapor}=\frac{R_{universal}}{mmo{l}_{vapor}}=\frac{10.73}{18.015}$ = 0.5956$\frac{psia\cdot f{t}^3}{lb\cdot R}$

$R_{air}$ is the specific gas constant of dry-air

$R_{vapor}$ is the specific gas constant water vapor, or 0.5956$\frac{psia\cdot f{t}^3}{lb\cdot R}$

$T_{rankine}=T_F+459.67$

${\rho }_{air}=\frac{P_{dry-air}}{0.3704\times (T_{DegF}+459.67)}+\frac{P_v}{0.5956\times (T_{DegF}+459.67)}$

$P_{dry-air}$ and $P_{vapor}$ are calculated in Specific Humidity Ratio

$CFM$ as calculated here

Alternatively, with estimated air density:

${\rho }_{air}\approx 0.075lb/f{t}^3$

$Q_{total-estimate}=4.5\times CFM\times (h_{return}-h_{supply})$

Sensible and Latent capacity

$Q_{sensible}={\rho }_{air}\times C_{pa}\times CFM\times 60min/hr\times (T_{return}-T_{supply})$

$C_{pa}$ is the specific heat of air, or $0.240\frac{Btu}{lb\cdot DegF}$

${\rho }_{air}$ can be used as calculated or estimated above

With estimated air density (and commonly found):

$Q_{sensible}=1.08\times CFM\times (T_{return}-T_{supply})$

$Q_{sensible}$ represents the energy used to effect the felt temperature change in the space/home.

The Latent Capacity is the difference between $Q_{total}$ and $Q_{sensible}$, this represents the amount of heat used to condense water out of the water/air-vapor mixture.

Sensible Heat Ratio

$SHR=Q_{sensible}/Q_{total}$

The $SHR$ will represent the efficiency of energy used for sensible cooling vs total cooling. Any given system has a mechanical and design limit to it’s cooling capacity, and then another practical limit based on the expected level of humidity.

Typical $SHR$ values can be found for certain loads and environment. Loads include things like equipment (pool, kitchen boiling water) and people (who are often the largest non-environmental contributor to indoor humidity). Environment is primarily the location of the space being conditioned. For example, home on the southeast coast will have a much higher latent load due to high humidity, than a home in arid Arizona.

$SHR$ for Charlotte, NC might be around 0.75

Static Pressure Sensing

Assuming static pressure 0.15-0.25 in. WC: Sea-level vs Actual = 1.28% error Elevation-adjusted vs Actual = 0.15% error

This might not be worth the extra sensor cost. More value to the purchase of the sensors would come from indication of clogged filters, which would reduced airflow and thus impact capacity or potentially damage a blower motor, as these motors can be over-taxed if under high static load.

In these cases, a $5-10sensorcouldsaveyoua$500+ on a visit from a technician and them replacing your filter or your burnt out blower motor.

Air Density Calculation

${\rho }_{air}=\frac{P_{dry-air}}{0.3704\times (T_{DegF}+459.67)}+\frac{P_v}{0.5956\times (T_{DegF}+459.67)}$

Air density is impacted by the temperature of the air, the humidity of that air through $P_v$ and the pressure of the air (which can be impacted greatly by elevation).

Using a constant of air density can allow variations in these factors to stack up to larger impacts on the real air density and thus on the final capacity. In the worst case hot and humid air capacity error can add up to 9%. This might be small, but all the sensors are present to calculate actual density and these errors can compound with others.

Predicting Desired SHR

Contact & Bypass Factor

The Contact and Bypass factors refer to the ratio of air that comes into contact with the coil, and is thus impacted by the cooling (or heating) affect of the coil. The Bypass Factor refers to the air that does not come into contact with the coil.

$BF=1-CF$

Apparatus Dew Point (ADP)

The Apparatus Dew Point is the theoretical dew point if the coil perfectly contacted all the air and pulled it down to the saturation temperature and condensed the water out of the air.

Ideal SHR

$SH{R}_{theoretical}=(h_{supply}-h_{ADP})/(h_{return}-h_{ADP})$

$h_{ADP}$ would be calculated using saturation pressures

Refrigerant Enthalpy

$T_{superheat}=T_{suction}-T_{saturated-suction}$

$T_{saturated-suction}$ comes from converting $P_{suction}$ using the refrigerant conversion table.

Refrigerant Saturation Temperature

When at the Saturation temperature, the refrigerant will be in the process of changing state, and partially liquid, partially gas. The temperature won’t change until the refrigerant has all changed to all one state or another (theoretically).

Superheat

Superheat is how many degrees beyond the saturation temperature the refrigerant is. This ensures that all of the refrigerant is turned into a gas before it reaches the compressor. In addition to this, many machines also include an accumulator, which can help to collect any potential liquid in the refrigerant and ensure it’s all evaporated before it gets to the compressor.

Superheating Value	What it means
$T<5DegF$	Overcharged, risk of liquid in the compressor
$5<T<12DegF$	Healthy
$15<T<25DegF$	Undercharged, or restricted airflow
$T>30DegF$	Very undercharged, or blocked filter

Subcool

$T_{subcool}=T_{saturated-liquid}-T_{liquid}$

$T_{saturated-liquid}$ comes from converting $P_{liquid}$ using the refrigerant conversion table. You can use $P_{discharge}$, however, this might be off by 1-2 psi compared to the actual suction pressure.

Subcool indicates how many degrees below its saturation temperature. Temperatures below this ensure that the refrigerant is fully liquid.

Charge Balancing

The Superheat and Subcool can be used to balance refrigerant charging. Low Subcool and High Superheat indicates an undercharged system. High Subcool and Low Superheat indicates a overcharged system.

Compressor Efficiency

Compressor Enthalpy reference

$\Delta H=Q-W$

$\Delta H$ is the change in enthalpy

$Q$ is the heat added to the system, for an ideal (adiabatic) compressor this is 0

$W$ is the work done on the system

In this ideal case, all work done by the compressor results in enthalpy change to the refrigerant.

${\eta }_{isentropic}=\frac{h_{discharge-ideal}-h_{suction}}{h_{discharge-actual}-h_{suction}}$

$h_{discharge-ideal}$ uses the $P_{discharge}$ and $S_{suction}$ and refrigerant properties

$S_{location}$ is the refrigerant entropy at a location, determined using refrigerant properties, $P_{location}$ and $T_{location}$

$h_{discharge-actual}$ is determined using refrigerant properties, $P_{discharge}$ and $T_{discharge}$

$h_{suction}$ is determined using refrigerant properties, $P_{suction}$ and $T_{suction}$

$S$ and $H$ refer to entropy and enthalpy respectively

Here are are calculating compressor efficiency by comparing the actual enthalpy change across the compressor and comparing that to the real enthalpy. In an ideal scenario there is no entropy increase, all that would occur is a change in enthalpy. Assuming this is the case, the ideal discharge enthalpy should calculate from the suction entropy and measured discharge pressure.

This efficiency number is typically specified by the compressor manufacturer and will never be perfect. The imperfection will show up as heat added to the refrigerant during operation (or expelled externally).

Finally, when doing compressor control on inverter units, this can be further used to control compressor efficiency when optimal cooling isn’t necessary.

Why Care?

The compressor efficiency should be close to the manufacturer specification at initial manufacture (ideally). This will drift with time as wear occurs on the motor and it’s efficiency decreases, maybe more friction is introduced with time.

In addition to a “gut check” on the compressor, you can reference degradation of other characteristics of the refrigerant loop to the compressor efficiency. If COP is reducing but compressor efficiency is staying steady, it can eliminate compressor degradation as a potential area of COP loss.

You can also use the steady degradation in isentropic efficiency to help determine when a compressor might begin to fail and create a risk profile over time for a customer.

Refrigerant Runtime Capacity

$\dot{m}=\frac{Q}{h_{out}-h_{in}}$

$Q_{evap}=\dot{m}\times (h_{suction}-h_{liquid})$

$Q_{compressor}=W_{compressor}\times {\eta }_{isentropic}$

$Q_{compressor}=\dot{m}\times (h_{discharge-actual}-h_{suction})$

$\dot{m}=W_{compressor}\times {\eta }_{isentropic}/(h_{discharge-actual}-h_{suction})$

$\dot{m}$ is the mass flow rate of the refrigerant and should be constant in a system with a single closed loop, replace one equation in the other.

$Q_{evap}=W_{compressor}\times {\eta }_{isentropic}/(h_{discharge-actual}-h_{suction})\times (h_{suction}-h_{liquid})$

$CO{P}_{refrigerant}=\frac{Q_{evap}}{W_{compressor}}$

$CO{P}_{refrigerant}=\frac{(h_{suction}-h_{liquid})\times {\eta }_{isentropic}}{h_{discharge-actual}-h_{suction}}$

This is another of calculation of tonnage based only on the refrigerant side of the system. This can be compared to the Airside tonnage.

Power Utilization

$I_{rms}=\frac{V_{rms}}{R_{burden}}\times C{T}_{ratio}$

$C{T}_{ratio}$ is defined by the data sheet

$W_{compressor}=V_{supply}\times I_{rms}\times PF$

$PF$ is power factor, without voltage measurement an estimated 0.9 can be used.

You can add any known power draw to determine the total system power use:

$W_{total}=W_{compressor}+W_{indoor}+W_{PCB-estimate}$

COP Calculation Conclusions

Differences in the below values can indicate a lose of efficiency or degradation of some subsystem. Other sensors can be evaluated to then diagnose a potential source of the issue.

Method	COP Calculation	Indepedent of
Air-side	$Q_{airside}/W_{compressor}$	Refrigerant-side
Refrigerant-side	$\frac{(h_{suction}-h_{liquid})\times {\eta }_{isentropic}}{h_{discharge-actual}-h_{suction}}$	Air-side & Electrical