First Law Accounting for Makers: Balancing Energy in Household Machines

What “First Law accounting” means in a workshop

For makers, the First Law is less about abstract theory and more about bookkeeping: you track energy crossing a chosen boundary and you reconcile it with what changes inside. The goal is to be able to look at a household machine (kettle, toaster, refrigerator, air compressor, dehumidifier) and answer: “Where did the input energy go?” and “How much of it became the effect I wanted?”

The First Law for a system is an energy balance. In maker terms, it is a checklist that prevents you from double-counting, missing a pathway, or blaming “mystery losses” that are actually stored energy or measurement error.

Choose the system, then write the balance

The most important decision is the boundary: what you include in the “system.” For household machines, two choices are common:

• Closed system (control mass): no mass crosses the boundary. Example: a sealed hot-water bottle cooling on a table.
• Open system (control volume): mass flows in/out. Example: a refrigerator (air and refrigerant circulate; heat crosses the walls; electrical power enters).

Most household machines are best treated as open systems because they move fluids (air, water, refrigerant) or because you care about inlet/outlet conditions (e.g., a fan heater warms air passing through).

General energy balance you will actually use

A practical form of the First Law for a control volume is:

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Rate form (control volume):  dE_cv/dt = Q̇_in - Q̇_out + Ẇ_in - Ẇ_out + Σ(ṁ_in e_in) - Σ(ṁ_out e_out)

Here, E_cv is the energy stored inside the control volume (internal + kinetic + potential). The term e is the specific energy carried by flowing mass:

e = h + (V^2)/2 + g z

For many household machines, kinetic and potential terms are small compared to thermal terms, so you often simplify to:

Common maker simplification:  dE_cv/dt ≈ Q̇_net + Ẇ_net + Σ(ṁ_in h_in) - Σ(ṁ_out h_out)

And for steady operation (after warm-up), stored energy stops changing:

Steady state:  dE_cv/dt = 0

Then the balance becomes a straightforward “in = out” accounting statement.

What counts as “energy in” and “energy out” in household machines

Electrical power is usually the easiest input to measure

For plug-in devices, electrical input power is often the dominant work term. You can measure it with a plug-in power meter (true RMS is best). Record:

• Instantaneous power (W)
• Energy over time (Wh or kWh)
• Duty cycle for cycling devices (fridges, dehumidifiers)

In First Law accounting, electrical power entering a device is typically treated as work input to the control volume. Internally it may become mechanical shaft work, resistive heating, or both.

Heat transfer is often the hardest term—so you infer it

Directly measuring heat transfer through walls is difficult without specialized sensors. Makers typically infer heat transfer by measuring changes in a fluid stream (air or water) or by measuring temperature rise of a known mass over time.

Two common inference routes:

• Calorimetry style: track temperature change of a known mass (e.g., water in a kettle) and compute energy stored in that mass.
• Flow heating/cooling: measure inlet/outlet temperatures and flow rate of air or water, then compute enthalpy change rate.

Stored energy matters during warm-up and cycling

Many devices have significant thermal mass (metal heat exchangers, compressor shell, toaster body). During warm-up, part of the input energy goes into raising the device’s own temperature. If you ignore stored energy, your “efficiency” estimate will look worse at startup and better later, and you may misinterpret what is happening.

For cycling devices, stored energy also appears as temperature swings in coils and housings. A good accounting separates:

• Transient phase: dE_cv/dt is not zero.
• Quasi-steady phase: average over a cycle so the net stored energy change over one full cycle is near zero.

Step-by-step workflow: First Law accounting on a real device

Step 1: Define the question and the boundary

Examples of clear questions:

• “How much of a kettle’s electrical energy ends up as increased energy of the water?”
• “How much heat does a small space heater deliver to the air stream?”
• “What is the heat removed from room air by a dehumidifier, and where does it go?”

Draw a simple box diagram. Label what crosses the boundary: electrical input, air in/out, water in/out, heat to room, condensate drain, etc.

Step 2: Decide whether you can assume steady state

If the device runs continuously and you wait for temperatures to stabilize, you can often use steady state. If it cycles, plan to average over multiple cycles. If it is a batch process (kettle heating water), it is inherently transient; you will track stored energy changes.

Step 3: Measure what you can directly

• Electrical: power meter (W, Wh), or clamp meter + voltage (be careful with mains safety).
• Temperatures: thermocouples, RTDs, or reliable digital probes placed to measure inlet/outlet streams (avoid measuring a hot surface when you need fluid temperature).
• Mass/flow: kitchen scale for batch mass, graduated cylinder for condensate, anemometer for air velocity (with duct area), or a simple timed fill for water flow.
• Time: log data or at least record timestamps.

Step 4: Convert measurements into energy rates or totals

Useful conversions for makers:

• Electrical energy: E_elec = ∫ P dt (or Wh reading × 3600 to get J).
• Sensible heating of a liquid (approx.): ΔE ≈ m c ΔT.
• Heating/cooling of a flowing stream (approx.): Q̇ ≈ ṁ c_p (T_out - T_in) for air/water when pressure changes are small.

When you use ṁ c_p ΔT, you are effectively using an enthalpy change approximation without needing full property tables, which is usually adequate for household conditions.

Step 5: Close the balance and compute the “unaccounted” term

After you compute the main terms, you will likely have a mismatch. Instead of calling it “loss,” label it as unaccounted and then investigate: is it heat leaking to the room, energy stored in the device, sensor placement error, or an ignored pathway (like latent heat in moisture)?

Write the balance with an explicit residual:

Residual = Inputs - Outputs - Storage change

A good experiment makes the residual small compared to the dominant terms, or at least explains it.

Example 1: Electric kettle as a batch energy balance

System and measurements

Choose the system as “water + kettle interior up to the waterline” or just the water, depending on what you want. If you choose only the water, then heat transfer from the heater to the water is internal to the device and you treat electrical energy as entering the water indirectly; the kettle body heating becomes part of the residual. A more honest maker boundary is: system = kettle + water, with electrical energy entering and heat leaving to the room.

Measurements:

• Mass of water m (kg) using a scale.
• Initial and final water temperature T1, T2.
• Electrical energy input E_elec from a plug-in meter.
• Optional: kettle exterior temperature rise (to estimate stored energy in the kettle body).

Step-by-step accounting

1) Fill kettle with a measured mass of water, e.g., m = 1.00 kg.

2) Measure initial water temperature, e.g., T1 = 20°C.

3) Start heating and record electrical energy until you stop at a target temperature (not necessarily boiling), e.g., T2 = 90°C. Suppose the meter reads E_elec = 0.090 kWh.

4) Convert electrical energy to joules:

E_elec = 0.090 kWh × 3.6×10^6 J/kWh = 324,000 J

5) Compute energy increase of the water (sensible):

ΔE_water ≈ m c ΔT ≈ (1.00 kg)(4180 J/kg·K)(90-20 K) = 292,600 J

6) The difference is what went into heating the kettle body and escaping to the room during the heat-up:

Residual ≈ E_elec - ΔE_water ≈ 324,000 - 292,600 = 31,400 J

7) Interpret the residual. Over a few minutes, 31 kJ is plausible as a combination of:

• Heating the metal/plastic kettle parts
• Heat loss from the kettle surface to surrounding air
• Heat carried away by steam if near boiling

8) Improve the accounting if desired: run the same test but stop at a lower temperature (e.g., 60°C) to reduce steam effects; or measure kettle mass and approximate its temperature rise to estimate stored energy in the kettle body.

What this teaches

This is First Law accounting in its simplest maker form: electrical energy in equals increased stored energy of water plus “everything else.” You are not trying to eliminate “everything else”; you are trying to quantify it and understand what design choices (insulation, lid use, heating rate) change it.

Example 2: Toaster as a steady-ish balance with strong heat losses

Why toasters are tricky

A toaster’s purpose is not to heat air efficiently; it is to deliver radiant and convective heat to bread surfaces and drive moisture out. Most of the electrical input ends up as heat in the room anyway, but the “useful” part is localized heating and drying of the bread.

Practical accounting approach

Define the system as “toaster + bread during one toast cycle.” Electrical energy enters. Outputs include:

• Increased energy of the bread (warming)
• Latent energy associated with moisture driven off (water evaporated)
• Heat released to the room air and surrounding surfaces

Step-by-step:

1) Weigh bread before and after toasting to estimate water mass lost: Δm_water.

2) Measure electrical energy for the cycle: E_elec.

3) Measure bread temperature rise roughly (surface IR is misleading; a probe in the crumb is better). Estimate sensible energy increase: m_bread c_bread ΔT (use an approximate c_bread around 2500–3000 J/kg·K depending on moisture).

4) Estimate evaporation energy: E_evap ≈ Δm_water × h_fg (latent heat of vaporization; use a typical value around 2.3–2.5 MJ/kg near kitchen temperatures).

5) The remainder is heat to the room and heating of the toaster body.

This example shows why “efficiency” must be defined carefully. If your definition of useful output is “energy absorbed by bread,” a toaster looks inefficient. If your definition is “browning and drying performance per Wh,” then the same accounting helps you compare settings, bread thickness, or reflective shields.

Example 3: Refrigerator as an open-system balance (and why the room warms)

Two heat transfers and one electrical input

A refrigerator moves heat from the cold interior to the warmer room using electrical power. For a steady operating period, the key accounting statement is:

Heat rejected to room = Heat removed from fridge interior + Electrical power input

In rate form:

Q̇_out,room ≈ Q̇_in,from_cold + Ẇ_elec

This is a First Law result: the room-side coil must dump not only the heat extracted from inside, but also the electrical work that ultimately becomes heat.

Maker measurement strategy

Directly measuring Q̇ at the coils is hard without airflow and temperature measurements. A practical maker method is to do a “thermal load test” inside the fridge:

• Place a known mass of water (sealed bottles) inside as a controllable thermal load.
• Measure water temperature change over time and the electrical energy consumed.

Step-by-step:

1) Put m = 2 kg of water in sealed bottles in the fridge. Let it equilibrate to fridge temperature.

2) Briefly warm the bottles outside to a known temperature (or add a known amount of warm water), then return them and track cooling back down. Measure electrical energy used during this recovery.

3) Estimate heat removed from the water:

Q_removed ≈ m c (T_start - T_end)

4) Compare Q_removed to electrical energy input during the same period. The ratio is not “efficiency” in the usual sense, but it is a performance indicator for the machine under those conditions.

5) Use the First Law statement to reason about the room: if the fridge draws 100 W on average and removes 150 W from inside, the room receives about 250 W while the compressor runs (averaged over time, scaled by duty cycle).

Example 4: Dehumidifier—where latent heat changes the accounting

Why dehumidifiers feel like heaters

A dehumidifier removes water vapor from air by cooling air below its dew point on an evaporator coil, condensing water, then reheating the air on a condenser coil. Electrically, it is similar to a small refrigerator, but both coils are in the same room. First Law accounting explains why the outlet air is often warmer than inlet air even though moisture is removed.

Boundary and terms

Choose the system as the whole dehumidifier. Inputs/outputs:

• Electrical work input Ẇ_elec
• Moist air in/out (mass flow carrying enthalpy)
• Liquid water out (condensate)
• Heat exchange with room air occurs via the air stream itself

A practical accounting focuses on two measurable outputs:

• Condensate rate (kg/s) from collected water over time
• Air temperature change across the unit (inlet/outlet)

Step-by-step maker test

1) Measure electrical power and energy over a 30–60 minute run.

2) Collect condensate in a container for a known time. Convert volume to mass (1 liter ≈ 1 kg).

3) Measure inlet and outlet air temperatures (place probes in the airflow, shielded from coil radiation).

4) Estimate latent heat removed from the air due to condensation:

Q̇_latent ≈ ṁ_condensate × h_fg

5) Estimate sensible heating of the outlet air if you can estimate airflow ṁ_air (from fan specs or anemometer):

Q̇_sensible ≈ ṁ_air c_p,air (T_out - T_in)

6) Use the First Law logic: the unit rejects to the room approximately the latent heat it removed (released during condensation on the cold coil) plus the electrical input, and it may also shift sensible heat. The net effect is usually room warming.

This is a strong example of why you must include phase-change energy when moisture is involved; otherwise the balance will not close and the device behavior will seem paradoxical.

Common pitfalls and how to avoid them

Mixing up “device efficiency” with “process effectiveness”

First Law accounting tells you where energy goes; it does not automatically tell you whether the device is “good.” You must define the desired output. For a kettle, useful output might be energy stored in water. For a dehumidifier, useful output might be kilograms of water removed per kWh. For a refrigerator, useful output might be heat removed from the interior per kWh under a given load.

Ignoring warm-up and thermal mass

If you measure a space heater for only the first minute, much of the input goes into heating the heater body. Average over a longer interval or explicitly include stored energy change by tracking body temperature rise (even approximately).

Bad temperature measurement locations

Measuring near hot surfaces can bias readings. For air streams, place the sensor in the moving air, away from radiant heaters and metal walls. For liquids, stir or allow time for mixing before reading.

Forgetting the “hidden” mass flow

Some devices move mass you might not notice: a vented dryer exhausts warm moist air; a range hood removes conditioned air from the house; a humidifier adds water vapor mass to air. If mass crosses the boundary, open-system terms matter.

Not checking units and magnitudes

Do quick sanity checks:

• 1 kWh = 3.6 MJ
• Heating 1 kg of water by 1°C takes about 4.2 kJ
• Evaporating 1 kg of water takes about 2.4 MJ

If your computed “useful heat” exceeds electrical input for a resistive heater, something is wrong. If your latent term is tiny but you collected a lot of water, something is wrong.

A reusable “maker template” for First Law energy audits

Template for batch devices (no mass flow you care about)

Given: electrical energy input E_elec over time interval Δt (from meter) Compute: change in stored energy of target material(s) ΔE_target ≈ Σ(m c ΔT) Optional: estimate stored energy in device body ΔE_device ≈ m_device c_device ΔT_device Residual (heat to room + unmodeled effects): E_res = E_elec - (ΔE_target + ΔE_device)

Template for flow devices (air or water through)

Given: electrical power P_elec, mass flow ṁ, inlet/outlet temperatures T_in, T_out Compute: thermal power to stream Q̇_stream ≈ ṁ c_p (T_out - T_in) If phase change: add Q̇_latent ≈ ṁ_phase × h_fg Residual: P_elec - (Q̇_stream + Q̇_latent) = net heat leak/storage (sign depends on boundary)

Template for cycling devices (fridge, dehumidifier)

Average over a full cycle or multiple cycles:

Over one full cycle:  ΔE_storage ≈ 0  so  Inputs ≈ Outputs (on average) Use time-averaged power and time-averaged flow/temperature data

With these templates, you can treat household machines as energy systems you can measure, model, and improve—by insulation, airflow management, duty-cycle control, or better operating practices—while keeping your accounting grounded in the First Law.