Theory and calculations review

Theo's diagnostic test had shown a pattern Dana had seen in many experienced apprentices: he understood the theory well enough to use it, but he made mistakes when the exam asked him to calculate it. A subcooling question answered with the wrong saturation temperature, a power question with the wrong formula for three-phase, a pressure question in gauge when it asked for absolute. Each mistake cost one mark, and together they had cost him nearly a tenth of his score. This lesson reviews the theory behind the calculations the exam uses most and sets out a short list of formulas worth knowing cold.

Kentucky's exam allows its code books, but the NATE exams supply little reference information, and a candidate who has to work out every formula loses time and confidence. More important, the formulas only help a candidate who knows when each one applies. That is what this lesson drills.

Pressure and temperature

Every refrigeration calculation starts from pressure and temperature. Gauge pressure is measured relative to the atmosphere; absolute pressure includes it. At sea level, absolute pressure is gauge pressure plus about 14.7 psi. Pressure-temperature charts and many formulas use absolute pressure, and compression ratio is always a ratio of absolute pressures, so a candidate who divides gauge pressures gets a ratio that is badly wrong, as RAC-110 and RAC-210 showed. Vacuum is measured from the atmosphere downward in inches of mercury on older gauges, and in microns of mercury or pascals absolute for evacuation.

The pressure-temperature relationship holds only where liquid and vapor are both present: in the evaporator while the refrigerant boils, in the condenser while it condenses, and in a cylinder with liquid in it. Superheated vapor and subcooled liquid are not at saturation, which is why superheat and subcooling are measured as differences from the saturation temperature at the measured pressure. For a single-component refrigerant or an azeotrope, one temperature corresponds to each pressure. For a blend with glide, there are two: the dew point, where the last liquid boils, and the bubble point, where the first vapor forms.

Superheat is the suction line temperature minus the evaporator saturation temperature, using the dew point for a blend. Subcooling is the condenser saturation temperature minus the liquid line temperature, using the bubble point for a blend. The two most common errors on the exam are using the wrong saturation point for a blend and subtracting in the wrong direction; both give answers that appear among the distractors.

Heat, energy and capacity

Sensible heat changes temperature; latent heat changes state. The heat to warm or cool a substance is its mass times its specific heat times the temperature change. The heat to change state is the mass times the latent heat. Refrigeration works because evaporation absorbs a great deal of latent heat at a low temperature and condensation releases it at a higher one, as RAC-110 explained.

For air, the field formulas use the heat capacity of a volume flow. Sensible heat in Btu per hour is about 1.08 times the airflow in cubic feet per minute times the temperature difference in degrees Fahrenheit; latent heat is about 0.68 times cfm times the humidity ratio difference in grains per pound, and total heat about 4.5 times cfm times the enthalpy difference in Btu per pound. For water, the factor is larger: heat in Btu per hour is about 500 times the flow in gallons per minute times the temperature difference in degrees Fahrenheit. The factors include standard air or water density, so they are close but not exact in hot, cold or high places.

Capacity units must be handled carefully. One ton of refrigeration is 12,000 Btu per hour, or about 3.517 kW. One kilowatt is about 3,412 Btu per hour. Efficiency is expressed as coefficient of performance, a ratio of heat moved to work input, or as energy efficiency ratio, Btu per hour per watt, or as seasonal ratings such as SEER2 and HSPF2, as RAC-320 described. EER equals COP times about 3.41.

Electrical calculations

Ohm's law, voltage equals current times resistance, applies to resistive loads and to circuit checks. Power in a resistive circuit is voltage times current. In an AC circuit with motors, real power is voltage times current times power factor for single-phase, and the square root of three, about 1.732, times line voltage times line current times power factor for three-phase. Forgetting the 1.732 is one of the commonest errors on the exam, and the answer that results is always among the options.

Resistances in series add; in parallel, the reciprocal of the total is the sum of the reciprocals, so two equal resistors in parallel have half the resistance of one. Capacitors work the other way: in parallel their capacitance adds; in series it falls. Transformer secondary current is the VA rating divided by the secondary voltage, so a 40 VA transformer at 24 V can supply about 1.7 A. Voltage drop in a conductor is current times the conductor's resistance there and back. Voltage unbalance on three-phase supplies is the largest deviation from the average voltage divided by the average, as RAC-130 showed.

Psychrometrics, loads and fluids

Psychrometric questions on the exam are mostly about reading and interpreting rather than complex calculation: what relative humidity means, why dew point matters, what happens to relative humidity when air is heated without adding moisture, how a cooling coil removes both sensible and latent heat, what sensible heat ratio means. RAC-330 covered the chart. Load questions are about the components of a load: transmission through walls, roof and windows, infiltration and ventilation, solar gain, people, lights and equipment, product loads in refrigeration, and the effect of each on equipment selection.

Fluid flow questions use the fan and pump laws: flow varies with speed, pressure with the square of speed and power with the cube of speed. A fan slowed to 80 % of its speed moves 80 % of the air at 64 % of the pressure with about 51 % of the power. Those laws explain why variable-speed drives save so much energy, and they appear in many forms on the exam.

A daily calculation drill

Dana's remedy for calculation slips was repetition with variety. Every evening session started with six mixed calculations, generated so that the numbers changed but the types did not, with the answers printed at the bottom to check immediately. The script is a seeded version, so the same set appears every time it runs with the same seed, which makes it useful for checking a method. Theo used a new seed each evening.

# A seeded mixed calculation drill with an answer key (same questions every time for seed 430)
import random
rng = random.Random(430)
qs = []
v, a = rng.choice([208, 230, 240]), rng.choice([12.5, 16.0, 18.4])
qs.append((f"Single-phase compressor at {v} V drawing {a} A, power factor 0.85: watts?", f"{v * a * 0.85:,.0f} W"))
amps, volts = rng.choice([3.2, 4.1, 5.0]), rng.choice([370, 389, 410])
qs.append((f"Start winding {amps} A, {volts} V across the run capacitor: microfarads?", f"{2652 * amps / volts:.1f} uF"))
line, sat = rng.choice([46, 50, 54]), rng.choice([19, 25, 36])
qs.append((f"Suction line {line} F, evaporator saturation {sat} F: superheat?", f"{line - sat} F"))
sat2, liq = rng.choice([100, 108, 115]), rng.choice([88, 93, 97])
qs.append((f"Condensing saturation {sat2} F, liquid line {liq} F: subcooling?", f"{sat2 - liq} F"))
cfm, dt = rng.choice([850, 1000, 1200]), rng.choice([18, 20, 22])
qs.append((f"{cfm:,} cfm across a coil, temperature drop {dt} F: sensible Btu/h (1.08 factor)?", f"{1.08 * cfm * dt:,.0f} Btu/h"))
psig = rng.choice([50, 90, 205])
qs.append((f"{psig} psig: absolute pressure at sea level?", f"{psig + 14.7:.1f} psia"))
for i, (q, ans) in enumerate(qs, 1):
    print(f"{i}. {q}")
print("answers:", " | ".join(f"{i}) {a}" for i, (_, a) in enumerate(qs, 1)))

1. Single-phase compressor at 230 V drawing 18.4 A, power factor 0.85: watts?
2. Start winding 3.2 A, 370 V across the run capacitor: microfarads?
3. Suction line 46 F, evaporator saturation 19 F: superheat?
4. Condensing saturation 108 F, liquid line 93 F: subcooling?
5. 1,000 cfm across a coil, temperature drop 18 F: sensible Btu/h (1.08 factor)?
6. 50 psig: absolute pressure at sea level?
answers: 1) 3,597 W | 2) 22.9 uF | 3) 27 F | 4) 15 F | 5) 19,440 Btu/h | 6) 64.7 psia

The six questions are the six types Theo missed most often: single-phase power with power factor, run capacitance, superheat, subcooling, sensible heat from airflow and absolute pressure. Each takes under a minute when the formula is known and the units are right. After three weeks of daily drills, his calculation errors on practice sets fell from about one in four to about one in fifteen.

Theo also learned to estimate before calculating. A 3-ton system is about 10 kW. A run capacitor for a small compressor is a few tens of microfarads. Superheat on a TXV system is usually a few kelvin. If a calculated answer is wildly different from the estimate, the calculation is probably wrong, and checking it is quicker than defending it. On the exam, an estimate also eliminates options that cannot be right before any arithmetic is done.

Reading charts and tables

Not every calculation on the exam uses a formula. Many use a chart or table: a pressure-temperature chart, a manufacturer's charging chart, a psychrometric chart, a superheat target table, a wire ampacity table or a capacity table. The skill is reading the right row and column, interpolating between values when needed and using the units the chart uses. Interpolation is a simple proportion: if a chart gives one value at 30 and another at 40, the value at 34 is 40 % of the way from the first to the second. Exam questions that give a short excerpt of a chart are testing exactly that skill.

Charts also have conditions printed on them. A charging chart is valid for a stated indoor wet bulb and outdoor dry bulb range; a capacity table for a stated airflow; a psychrometric chart for a stated pressure, usually sea level. A candidate who reads a value from the chart without checking the conditions can get an answer that is off by a wide margin, and the question writer may have included that wrong value as a distractor.

Units and conversions

US practice uses imperial units, but the trade meets SI units on imported equipment, in some manufacturers' literature and in the codes' metric equivalents. The conversions worth knowing are few: one ton is 12,000 Btu per hour and about 3.517 kW; one kilowatt is about 3,412 Btu per hour; one psi is about 6.895 kPa; one inch of water column is about 249 Pa; and a temperature difference of one degree Fahrenheit equals 5/9 of a kelvin. A temperature itself converts as degrees Fahrenheit equal 1.8 times degrees Celsius plus 32, but a difference converts by the factor alone, and confusing the two is a common slip.

Writing units at every step of a calculation is the simplest defense against unit errors. It looks slow, but it is faster than finding the error afterwards, and it turns many wrong answers into obviously wrong ones before they are chosen.

Dana added one habit for the exam room: before calculating, glance at the options. If they differ by a factor of ten, a rough estimate is enough to choose; if they differ by a few percent, the full calculation is needed, carefully. The options tell a candidate how precise the answer has to be.