3.1 Energy transfer by heat, work, and mass

Energy transfer is the heart of thermodynamics. Heat, work, and mass transfer are the three ways energy crosses a system boundary. Understanding how each one works, and how to account for them quantitatively, is what the First Law is all about.

The First Law of Thermodynamics says energy can't be created or destroyed, only transferred or converted. This gives you a bookkeeping tool: if you can track the heat, work, and mass crossing a system boundary, you can figure out how the system's energy changes.

Heat, Work, and Mass Transfer

Defining Heat, Work, and Mass Transfer

Heat ($Q$) is energy transfer driven by a temperature difference. It always flows spontaneously from higher temperature to lower temperature. No temperature difference, no heat transfer.

Work ($W$) is energy transfer driven by a force acting through a displacement, or more generally, any energy transfer that isn't heat or mass flow. In thermodynamics, the most common form is boundary work, where a gas expands or compresses against a piston.

Mass transfer moves matter across a system boundary, and that matter carries energy with it (internal energy, kinetic energy, potential energy, and flow work bundled together as enthalpy). This only applies to open systems (also called control volumes). Closed systems have no mass crossing the boundary.

A quick way to keep them straight:

• Heat transfer depends on temperature differences
• Work depends on force and displacement (or equivalent interactions like shaft work or electrical work)
• Mass transfer involves the physical movement of matter across the boundary

Energy Transfer Impact on System Properties

Each mode of energy transfer changes different system properties:

• Heat transfer most directly changes a system's temperature (and internal energy). Add heat to a gas at constant volume and its temperature and pressure both rise.
• Work often shows up as changes in volume or pressure. Compress a gas and you do work on it, raising its pressure and temperature.
• Mass transfer changes the total energy and possibly the composition of the system. Fluid flowing into a turbine carries enthalpy in; fluid flowing out carries less.

Heat itself reaches a system through three physical mechanisms: conduction, convection, and radiation. These are covered in detail in the next section.

Mechanisms of Energy Transfer

Conduction

Conduction transfers heat through direct molecular contact, with no bulk motion of the material. Higher-energy (hotter) particles collide with lower-energy (cooler) neighbors, passing kinetic energy along. Think of a metal spoon sitting in hot coffee: energy travels up the spoon from molecule to molecule.

The rate of conduction is governed by Fourier's law:

$\dot{Q}_{cond} = -kA \frac{dT}{dx}$

where:

• $\dot{Q}_{cond}$ = rate of heat transfer (W)
• $k$ = thermal conductivity of the material (W/m·K)
• $A$ = cross-sectional area perpendicular to heat flow (m²)
• $\frac{dT}{dx}$ = temperature gradient (K/m)

The negative sign means heat flows in the direction of decreasing temperature.

Thermal conductivity ($k$) is a material property. Metals like copper ($k \approx 400$ W/m·K) conduct heat very well. Insulating materials like fiberglass ($k \approx 0.04$ W/m·K) resist heat flow. This is why insulation works: low $k$ means a small $\dot{Q}$ even with a large temperature difference.

Convection

Convection transfers heat through the bulk movement of a fluid (liquid or gas). It combines conduction at the surface with fluid motion that carries energy away.

Two types:

• Natural (free) convection: Fluid motion is driven by buoyancy. Hotter fluid is less dense and rises; cooler fluid sinks to replace it. Example: warm air rising from a radiator.
• Forced convection: Fluid motion is driven by an external device like a fan, pump, or blower. Example: a car radiator with a fan pushing air across the fins.

Forced convection generally transfers heat much faster because the fluid velocity is higher.

The rate of convective heat transfer is described by Newton's law of cooling:

$\dot{Q}_{conv} = hA(T_s - T_\infty)$

where:

• $h$ = convective heat transfer coefficient (W/m²·K)
• $A$ = surface area (m²)
• $T_s$ = surface temperature (K)
• $T_\infty$ = bulk fluid temperature far from the surface (K)

The value of $h$ depends on the fluid, flow conditions, and geometry. It's not a simple material property like $k$. Typical values range from about 5–25 W/m²·K for natural convection in air up to 10,000+ W/m²·K for forced convection with liquid water.

Radiation

Radiation transfers energy through electromagnetic waves. Unlike conduction and convection, it requires no medium and can travel through a vacuum (this is how the Sun heats the Earth).

Every object above absolute zero emits thermal radiation. The rate of emission is given by the Stefan-Boltzmann law:

$\dot{Q}_{rad} = \varepsilon \sigma A T^4$

where:

• $\varepsilon$ = surface emissivity (dimensionless, 0 to 1)
• $\sigma$ = Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W/m²·K⁴)
• $A$ = surface area (m²)
• $T$ = absolute surface temperature (K)

Notice the $T^4$ dependence. This means radiation becomes dominant at high temperatures. At 300 K (room temperature), radiation is modest. At 1500 K (a furnace), it's enormous.

Emissivity ($\varepsilon$) ranges from 0 (perfect reflector) to 1 (perfect emitter, called a blackbody). Polished metals have low emissivity ($\varepsilon \approx 0.05$), while rough, dark surfaces approach $\varepsilon = 1$.

Energy Changes in Thermodynamic Processes

First Law of Thermodynamics and Energy Balance

The First Law in equation form for a closed system (no mass crossing the boundary):

$\Delta U = Q - W$

• $\Delta U$ = change in internal energy of the system
• $Q$ = net heat added to the system
• $W$ = net work done by the system

For an open system (mass crosses the boundary), the energy balance in steady-state rate form is:

$\dot{Q} - \dot{W} = \sum \dot{m}_{out} h_{out} - \sum \dot{m}_{in} h_{in}$

where $\dot{m}$ is the mass flow rate and $h$ is specific enthalpy. (Kinetic and potential energy terms are often included too, but they're frequently negligible.) The key idea: mass flowing in carries enthalpy with it, and mass flowing out takes enthalpy away.

Thermodynamic Processes and Energy Changes

Four idealized processes show up constantly. For each one, a constraint simplifies the First Law:

• In an isothermal process for an ideal gas, internal energy depends only on temperature, so $\Delta U = 0$. All the heat added goes directly into work output.
• In an isochoric process, the volume doesn't change, so no boundary work is done. All heat goes into changing internal energy.
• In an adiabatic process, the system is perfectly insulated (or the process is fast enough that heat doesn't have time to transfer). Any work done comes at the expense of internal energy, so the temperature changes.
• $\gamma = c_p / c_v$ is the specific heat ratio (about 1.4 for air at room temperature).

Problem Solving for Heat, Work, and Mass Transfer

Applying Energy Balance Equations

Here's a systematic approach for energy balance problems:

1. Define the system. Draw a boundary. Decide if it's closed or open.

2. Identify the process. Is it isothermal, isobaric, isochoric, adiabatic, or something else?

3. Write the appropriate energy balance.

   • Closed: $\Delta U = Q - W$
   • Open (steady-state): $\dot{Q} - \dot{W} = \sum \dot{m}_{out} h_{out} - \sum \dot{m}_{in} h_{in}$

4. Apply the constraint to simplify. For example, if $Q = 0$ (adiabatic), cross out the $Q$ term.

5. Substitute known values and solve for the unknown.

6. Check signs. Does the direction of heat/work make physical sense?

Using Process-Specific Equations

For an ideal gas, each process type gives you a shortcut:

• Isothermal (constant $T$): $PV = \text{constant}$, and work is $W = nRT \ln(V_2/V_1)$
• Isobaric (constant $P$): $W = P(V_2 - V_1)$, and $Q = mc_p \Delta T$
• Isochoric (constant $V$): $W = 0$, and $Q = mc_v \Delta T$
• Adiabatic ($Q = 0$): $PV^\gamma = \text{constant}$, and $T V^{\gamma-1} = \text{constant}$

When solving, always check whether the substance is actually behaving as an ideal gas. The ideal gas law ($PV = nRT$) works well at low pressures and high temperatures relative to the substance's critical point. For steam, refrigerants, or gases near their saturation conditions, you'll need property tables instead.

Applying Specific Heat Capacity and Ideal Gas Law

Specific heat capacity connects heat transfer to temperature change:

$Q = mc\Delta T$

where $m$ is mass, $c$ is specific heat, and $\Delta T = T_{final} - T_{initial}$.

For ideal gases, there are two specific heats:

• $c_v$ (constant volume): used when volume is fixed
• $c_p$ (constant pressure): used when pressure is fixed
• They're related by $c_p - c_v = R$ (on a per-mass basis, using the specific gas constant)

The ideal gas law ties the state variables together:

$PV = nRT$

This is your go-to equation for relating pressure, volume, and temperature when the gas behaves ideally. You can also write it as $Pv = RT$ using specific volume ($v = V/m$) and the specific gas constant.

Analyzing Thermodynamic Cycle Efficiency

A thermodynamic cycle returns the system to its initial state, so $\Delta U_{cycle} = 0$. That means:

$W_{net} = Q_{in} - Q_{out}$

Thermal efficiency is defined as:

$\eta = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$

For a Carnot cycle (the theoretical maximum efficiency between two reservoirs):

$\eta_{Carnot} = 1 - \frac{T_L}{T_H}$

where $T_L$ and $T_H$ are the absolute temperatures (in Kelvin) of the cold and hot reservoirs. No real engine can exceed this efficiency. For example, a Carnot engine operating between 300 K and 600 K has a maximum efficiency of $1 - 300/600 = 0.50$, or 50%.

Other cycles (Otto, Diesel, Rankine) have their own efficiency expressions, but they all follow the same core idea: efficiency equals net work out divided by total heat in.