Heat and Work as Energy Transfer: Sign Conventions and Boundaries

Why “heat” and “work” are not things you store

In maker projects it is common to say “the motor has work in it” or “the hot block contains heat.” In thermodynamics, that wording causes mistakes. Heat and work are not properties contained inside a device. They are modes of energy transfer across a boundary. A battery, flywheel, compressed spring, or hot reservoir can store energy in different forms, but “heat” and “work” describe how energy crosses from one place to another during a process.

This chapter focuses on two practical ideas you will use constantly when analyzing real machines: (1) choosing the system boundary and (2) applying sign conventions so that heat and work terms in your equations match the actual direction of energy transfer. Most confusion in thermodynamics for real devices comes from mixing boundaries and sign conventions.

System, surroundings, and the boundary you draw

A system is whatever you decide to analyze. Everything else is the surroundings. The boundary is the (real or imaginary) surface separating them. You can draw the boundary around a piston-cylinder, around a turbine, around a whole engine, or around a room. The boundary choice changes which energy transfers appear as heat or work and which are internal to the system.

Closed system vs control volume (open system)

• Closed system: no mass crosses the boundary. Energy can cross as heat and/or work. Example: a sealed pressure cooker (ignoring small leaks).
• Control volume (open system): mass can cross the boundary. Energy can cross with mass (flow) and also as heat and/or work. Example: an air compressor with inlet and outlet.

This chapter emphasizes heat and work as boundary interactions. For open systems, you still use the same sign conventions for heat and work, but you must be careful about energy carried by the flowing mass (often handled with enthalpy and kinetic/potential terms). The key point remains: heat and work are defined at the boundary.

Boundary placement changes what you call “work”

Consider a DC motor driving a fan. If your system is only the motor housing, then electrical energy crosses the boundary through wires and mechanical shaft work crosses the boundary through the shaft. If your system includes both motor and fan, then the shaft is internal and the mechanical transfer between motor and fan is no longer a boundary work term; instead, you would see electrical input and perhaps aerodynamic work leaving to the air as pressure/flow effects plus heat losses to the surroundings.

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For maker analysis, always sketch the boundary first. Then list every way energy crosses it: electrical, mechanical, thermal, and (if open) with mass flow.

Heat transfer: what qualifies as “heat”

Heat is energy transfer across the boundary because of a temperature difference. That is the defining criterion. If energy crosses due to a temperature gradient, it is heat transfer, regardless of whether the mechanism is conduction, convection, or radiation.

• Conduction: through solids or stationary fluids (e.g., heat leaking through an insulated wall).
• Convection: carried by moving fluid at the boundary (e.g., a heatsink with airflow).
• Radiation: electromagnetic emission/absorption (e.g., a glowing heating element warming a nearby surface).

In practice, you often do not need to separate these mechanisms in the first-law bookkeeping; you only need the net heat transfer rate Q̇ across the boundary. But you must ensure you are not accidentally counting energy transfer that is not driven by temperature difference (for example, electrical power in a wire is not heat transfer; it is electrical work crossing the boundary).

Heat is path-dependent

Heat transfer depends on the process path (how the system changes and how it is connected thermally). That is why we write heat as Q (a process quantity) rather than as a stored property. Two different heating methods can bring a system to the same final condition while transferring different amounts of heat because other interactions (work, losses) differ.

Work transfer: what qualifies as “work”

Work is energy transfer across the boundary by any mechanism other than temperature difference. In maker devices, common work modes include mechanical shaft work, electrical work, and boundary (expansion/compression) work.

Common work modes in real machines

• Shaft work: torque through a rotating shaft (motors, turbines, generators). Power is Ẇ_shaft = τ ω.
• Electrical work: energy transfer via electric potential and current crossing the boundary. Power is Ẇ_elec = V I (DC) or Ẇ = V_rms I_rms cosφ (AC real power).
• Boundary (p–V) work: moving boundary due to pressure forces, typical in piston-cylinder devices. Differential form for a quasi-equilibrium process: δW_b = p dV.
• Other work-like transfers: stirring (paddle wheel), stretching a membrane, magnetization work in special devices. In maker contexts these are less common but the same boundary logic applies.

Work is also path-dependent. For example, compressing air quickly vs slowly can require different work because heat transfer differs during the process.

Sign conventions: choose one and stick to it

Sign conventions are not “laws of nature”; they are bookkeeping rules. The danger is mixing conventions between sources or between closed-system and control-volume equations. The safest approach is to write the first law with a clearly stated sign convention and then apply it consistently.

Most common convention in engineering thermodynamics

• Heat transfer to the system is positive: Q > 0 when heat enters the system.
• Work done by the system is positive: W > 0 when the system does work on the surroundings.

With that convention, the closed-system first law is commonly written as:

ΔE_system = Q - W

Here ΔE_system is the change in the system’s total energy (internal + kinetic + potential, as needed). If the system does positive work on the surroundings, energy leaves the system, so it subtracts.

Alternative convention you will also see

Some fields (especially chemistry) use “work done on the system is positive.” Then the first law is written as ΔE = Q + W. This is not wrong, but it is different. If you use it, you must flip the sign of every work term compared to the engineering convention.

For maker work, the engineering convention (ΔE = Q - W) aligns well with machines: engines produce positive work output; compressors require negative work (work input).

A practical sign test: “does energy cross the boundary into the system?”

When you are unsure, do this: draw an arrow for each interaction across the boundary. If energy enters as heat, Q is positive. If energy leaves as heat, Q is negative. For work, decide whether the system is doing work on the surroundings (arrow out as work) or surroundings are doing work on the system (arrow in as work). Under the engineering convention, arrow out as work means W > 0; arrow in as work means W < 0.

Boundary work in piston-cylinder devices: the sign in your hands

Piston-cylinder setups are a clean way to see sign conventions because the work is literally force times displacement. Consider a gas in a cylinder pushing a piston outward.

Quasi-equilibrium boundary work

If the process is slow enough that the gas pressure at the boundary is well-defined and nearly uniform, the boundary work is:

W_b = ∫ p dV

• If the gas expands, dV > 0, so W_b > 0. The system does work on the surroundings.
• If the gas compresses, dV < 0, so W_b < 0. The surroundings do work on the system.

This matches intuition: expansion can drive a crank; compression requires you to push.

Step-by-step: compute boundary work from a simple process

Scenario: A piston-cylinder contains gas that expands at constant pressure p = 200 kPa from V1 = 0.010 m³ to V2 = 0.030 m³. Find boundary work using the engineering sign convention.

• Step 1: Identify the system and boundary. System = gas inside cylinder. Boundary includes piston face.
• Step 2: Identify the work mode. Moving boundary work (p–V work).
• Step 3: Write the work integral. For constant pressure: W_b = ∫ p dV = p (V2 - V1).
• Step 4: Substitute numbers. W_b = 200,000 Pa × (0.030 - 0.010) m³ = 200,000 × 0.020 = 4,000 J.
• Step 5: Apply sign. Expansion means V2 > V1, so W_b = +4.0 kJ.

If you later use ΔE = Q - W, this positive work will reduce the system energy unless enough heat enters to compensate.

Shaft work and electrical work: keep them separate from heat

In real maker builds, the most common mistake is to treat electrical input power as “heat added.” Electrical power crossing the boundary is a work term, even if it ultimately becomes internal energy and then heat loss. The classification depends on the boundary mechanism, not on what the energy “turns into” later.

Example: resistive heater inside a sealed box

Suppose you place a resistive heater inside a sealed, rigid, insulated box and power it from an external supply through wires passing the boundary.

• Electrical energy crosses the boundary through the wires: that is electrical work input to the system (under engineering convention, work done by the system is positive, so work input means W < 0).
• If the box is insulated, Q ≈ 0 (no heat transfer across the boundary).
• The gas and internal components warm up, so ΔE_system > 0.

With ΔE = Q - W, if Q = 0 and ΔE > 0, then -W > 0 so W < 0, consistent with work input.

If instead you draw the boundary around only the gas (excluding the heater element), then energy crosses the gas boundary as heat from the hot element to the gas (temperature difference), and you would model it as Q_gas > 0. Same physical reality, different boundary, different classification.

Heat loss vs useful work: what your boundary reveals

For makers, the boundary is a design tool: it helps you separate useful work output from losses. Consider a small air compressor driven by an electric motor.

• If your system is the entire compressor + motor assembly, then electrical power enters as work, compressed air leaves carrying energy with mass flow, and heat leaves through the casing.
• If your system is only the compressor control volume (excluding motor), then shaft work enters through the shaft, air flows in/out, and heat may leave.

Both are valid, but they answer different questions. The first helps you estimate wall-plug efficiency. The second helps you estimate compression work and discharge temperature behavior.

Step-by-step workflow for assigning signs in any problem

Use this checklist whenever you model a device. It prevents 90% of sign errors.

• Step 1: Draw the system boundary. Include a quick sketch. Mark inlets/outlets if it is open.
• Step 2: List all energy interactions across the boundary. Typical list: heat transfer Q, shaft work W_sh, electrical work W_el, boundary work W_b, and (for open systems) energy with mass flow.
• Step 3: Put arrows on each interaction. Arrow into system or out of system.
• Step 4: Choose the sign convention explicitly. In this course: Q positive into system; W positive out of system (done by system).
• Step 5: Assign signs from arrows. Heat arrow in: Q > 0. Heat arrow out: Q < 0. Work arrow out: W > 0. Work arrow in: W < 0.
• Step 6: Write the first-law balance. For a closed system: ΔE = Q - W. For a control volume, write an energy rate balance with Q̇ and Ẇ plus flow terms.
• Step 7: Sanity-check with limiting cases. If you remove insulation, should Q become negative (heat loss)? If you disconnect the shaft, should W_sh go to zero? If the device is a motor, should shaft work be leaving the system?

Common sign-convention pitfalls in maker projects

Mixing “work input” language with the symbol sign

People often say “work input is positive.” That is a verbal convention, not the engineering sign convention. Under ΔE = Q - W, work input corresponds to W < 0. To avoid confusion, label terms as W_in and W_out or keep the arrow diagram and let the sign convention handle it.

Counting the same transfer twice

If you include both electrical power into a motor and shaft power out, do not also add “heat generated by the motor” as an independent input. The heat generated inside the motor is not a boundary transfer; it is an internal conversion that will later leave as heat transfer through the casing (a boundary heat term) if your boundary includes the casing. Double counting happens when internal dissipation is treated as an extra boundary term.

Calling friction “heat transfer” across the boundary

Friction at a boundary can be tricky. If friction occurs at the boundary (for example, a brake pad rubbing a rotating disk and your system is the disk), the energy transfer across the boundary due to friction is classified as work (mechanical work), not heat, because it is not driven by a temperature difference. The frictional work then becomes internal energy and may later leave as heat transfer due to temperature difference.

Boundary selection examples you can reuse

Example A: hand-crank generator charging a battery

Draw the boundary around the generator only.

• Shaft work enters from your hand: work arrow into system, so W < 0.
• Electrical work leaves through wires to the battery: work arrow out, so W > 0 for that electrical term if you treat it as work done by the generator on the surroundings.
• Heat leaves from generator casing to air: Q < 0.

If instead you draw the boundary around generator + battery, the electrical transfer becomes internal, and you would see shaft work in and heat out, plus chemical energy storage changes inside the boundary.

Example B: insulated piston-cylinder with a paddle wheel

System = fluid in cylinder. Boundary is rigid except for paddle shaft seal (no volume change).

• Insulated: Q = 0.
• No boundary work from volume change: W_b = 0.
• Shaft work enters via paddle wheel: W_sh < 0.

Then ΔE = -W_sh, so the fluid’s energy increases due to stirring. This is a classic maker-relevant model for mixing viscous fluids where temperature rises even without a heater.

Heat and work rates vs totals: matching sensors to equations

In experiments you often measure power and temperature over time. Power is a rate: Q̇ or Ẇ. Total heat/work over a time interval is the time integral:

Q = ∫ Q̇ dt     W = ∫ Ẇ dt

Practical mapping to maker instrumentation:

• Electrical work rate: measure V and I to estimate Ẇ_elec.
• Shaft work rate: measure torque and speed (or infer from motor constants) to estimate Ẇ_shaft.
• Heat transfer rate: estimate from temperature difference and thermal resistance (or from calorimetry), but be careful: what you infer as “heat” must be across the boundary due to temperature difference.

When you integrate power over time, keep the sign consistent with your arrows. For example, if electrical power is entering your system, under the engineering convention it contributes negative work (W becomes more negative as time passes), even though the measured electrical power V I is a positive number. You can handle this cleanly by writing Ẇ = -V I for work into the system, or by defining separate Ẇ_in and Ẇ_out terms.

Quick reference: sign convention table (engineering)

• Heat into system: Q > 0
• Heat out of system: Q < 0
• Work done by system on surroundings (shaft out, expansion work out, electrical power delivered out): W > 0
• Work done on system by surroundings (shaft in, compression work in, electrical power supplied in): W < 0

Use the boundary sketch and arrows first, then translate arrows into signs using this table. That habit makes your thermodynamic accounting reliable across engines, compressors, heaters, generators, and hybrid maker contraptions.