Thermodynamics Basics: Systems, Boundaries, and PV Diagrams (Lecture Notes)

Administrative reminders from the lecture

SpaceX update: The speaker highlights a recent successful SpaceX launch of the Super Heavy rocket and notes the rapid, high-pressure work environment there; the rocket is extremely large and the landing splashdown is dramatic, approaching science-fiction scale in appearance.
A lighter aside mentions someone (SpaceX employee’s brother) officiating a ceremony; a human-interest moment amid the technical talk.
Academic honesty form: By Friday, download and sign the academic honesty signature form from the content section, then upload to the drop box. A simple one-page form acknowledging expectations.
Watch prerequisite videos: Scherzer’s lessons 1–3 and Degani’s video. The class will move quickly through definitions, so being prepared matters.
The lecture will introduce key thermo concepts: system boundaries, state vs. process, and the PV diagram foundation.

Key concepts: system, surroundings, control surface, and control volume

System: The portion of the universe being studied; defined by a boundary (control surface).
Surroundings: Everything outside the system.
Control surface: The boundary that separates the system from its surroundings; drawn as a red dotted line to denote the system.
Control volume (open system): A region where mass can cross the boundary; mass inflow and outflow are allowed.
Closed system (control mass): Mass cannot cross the boundary; mass is constant.
Open system (control volume): Mass can cross the boundary; mass flow in and out can occur.
Examples:
- Rigid tank: A closed system where the volume does not change with pressure/temperature changes.
- Piston-cylinder: A closed system where the boundary (piston) can move, changing the volume while mass stays constant.
- Nozzle/t turbine/pump: Examples of open systems where mass flows through and the boundary can move with the flow.
Schematic idea for a rigid tank problem: system inside a control surface; focus on how pressure, temperature, and density change without changing mass or volume for the rigid tank.

State, process, and the role of q and w in energy transfer

State: A snapshot defined by properties at a given time (e.g., pressure P1, temperature T1, volume V1, density ρ1).
Process: Any change that moves the system from an initial state (state 1) to a final state (state 2); a process updates the system’s properties.
State points vs properties: Properties at a state (state 1 or state 2) are state-point properties; e.g., P1, T1, V1, ρ1.
q (or Q): Heat transfer into the system; convention in this course is that heat transfer into the system is positive.
w (or W): Work transfer; convention in this course is that work leaving the system is positive (i.e., boundary work). Heat in is positive; work out is positive.
HIPTOIN mnemonic (helps remember sign conventions):
- Heat in is positive
- Positive work is defined as work out of the system
- Work in is negative; conversely, work out is positive
For energy balance in a closed system (no mass flow):
- First law in this context: $\boxed{ \frac{dU}{dt} = \frac{dQ}{dt} - \frac{dW}{dt} }$
- Integrated form for a process: oxed{ U = Q - W }
Sign convention recap:
- Heat transfer into the system: q > 0
- Work leaving the system: w > 0
- If work is negative, energy is entering via boundary work; if work is positive, energy is leaving via boundary work.
Historical note: The coal-engine era influenced the sign convention (heat added to drive the wheels vs. energy carried away by the exhaust) in shaping the current positive-sign convention for heat and for boundary work.
In this class, capital vs. lowercase notation matters: capital Q and W may be used in narrative; the core concepts use q, w, q̇ (heat rate), and ẇ (work rate).

Closed vs. open systems: mass transfer and control volume

Closed system (control mass): mass does not cross the control surface; mass is constant.
Open system (control volume): mass can cross the boundary; mass flow in and out is characterized by the mass flow rate, $\dot{m}$ .
Mass flow rate: $\dot{m} = \frac{dm}{dt}$ (kg/s in SI; lbm/s in English units).
For open systems, heat and work are often described as rates: $\dot{Q}$ (heat transfer rate) and $\dot{W}$ (work rate, energy per unit time).
Examples of open-system devices: nozzle (mass flows in/out with velocity changes), turbine, pump.
State variables for open systems include state properties at the inlet and outlet (e.g., P1, T1, V1, velocity V1; P2, T2, V2, velocity V2).
The importance of time: open-system devices involve flow and a time component; we encounter velocities and mass flow rates.

Property types: extensive, intensive, and specific properties

Extensive properties: depend on the size or extent of the system (e.g., mass, total volume, total energy U, total enthalpy H).
- Examples: mass (m), total volume (V), total internal energy (U).
Intensive properties: do not depend on system size (e.g., temperature T, pressure P, density ρ).
Specific properties: extensive property divided by mass; a normalization to allow comparisons across systems of different sizes.
- Definition: for any extensive property A, the corresponding specific property is $a = \frac{A}{m}$ .
- Common examples:
- Specific volume: $v = \frac{V}{m}$ (units: m^3/kg or ft^3/lbm)
- Specific internal energy: $u = \frac{U}{m}$
- Specific enthalpy: $h = \frac{H}{m}$
Why specific properties matter:
- Tables of thermodynamic properties tabulate specific properties; the total property for a system of mass m can be recovered by multiplying the specific property by mass: $A = m a$ .
- This normalization allows universal tables usable for any system once you know the mass.
Notation:
- Capital V and lowercase v distinguish total volume from specific volume; similarly U vs u, H vs h.
Typical units for specific properties (SI):
- Specific volume: $v \;\text{[m}^3\text{/kg]}$
- Specific internal energy: $u \;\text{[J/kg]}$
- Specific enthalpy: $h \;\text{[J/kg]}$
Practical takeaway:
- To go from a specific property to a total property, multiply by the mass: $A = m a$ ; to go from total to specific, divide by mass: $a = A/m$ .

Mass flow rate, energy rates, and common unit considerations

In open systems, mass flow rate, heat rate, and work rate are central:
- Mass flow rate: $\dot{m} = \frac{dm}{dt}$ (kg/s, lbm/s)
- Heat transfer rate: $\dot{Q}$ (W in SI; BTU/hr in English units; other units include kW, MW, etc.)
- Work rate: $\dot{W}$ (W in SI; horsepower, etc. in English units)
Relationship to power: $\dot{W} = \text{power}$ ; 1 W = 1 J/s; 1 HP ≈ 746 W.
Common English-English unit considerations:
- Mass: pound-mass (lbm)
- Force: pound-force (lbf)
- Energy: foot-pound force (ft-lbf) or BTU (1 BTU ≈ 1055 J)
- Power: horsepower (HP); 1 HP ≈ 745.7 W
In the English engineering unit system, the mixing of mass and force units requires care in conversions; the equation sheet is used for these conversions in problems.

Boundary work and the PV diagram foundation

Boundary work is the work associated with moving the system boundary (volume changes):
- Definition: $\boxed{ Wb = \int{V1}^{V2} P \, dV }$
- Conceptual origin: A piston with cross-sectional area A experiences a force F = P A; the piston displacement dx gives a volume change dV = A dx; energy transfer due to this boundary motion is the work term.
- In a process from state 1 to state 2, the boundary work is the work done by the system on its surroundings (area under the P-V curve).
PV diagram: two-dimensional plot with pressure P (vertical axis) and volume V (horizontal axis).
- Work is represented by the area under the P-V curve between states 1 and 2: $Wb = \int{V1}^{V2} P \; dV$
- Special case: If the process is isobaric (P constant):
- State 1 to State 2 with P1 = P2 = P
- The work becomes the rectangle area: $Wb = P (V2 - V_1)$
Process terminology on the PV diagram:
- A process is the path from state 1 to state 2 on the PV diagram, not just the end states.
- The subscript b on W denotes boundary work (as opposed to other forms, like shaft work).
For open systems, boundary work still applies to the moving boundary, but you also have mass flow work and possibly other forms (shaft work) depending on the device; the basic area under P-V remains a foundational concept for boundary interaction.
Important takeaway: Work is the area under the curve for a given process on a PV diagram; boundary work is specifically the energy transfer due to boundary movement, quantified by the integral of pressure with respect to volume.

Special process: isobaric (constant pressure) expansion example

Scenario: A closed, constant-pressure process where the boundary moves from volume V1 to V2 with P constant.
- State 1: (P1, V1) and State 2: (P2, V2) with P1 = P2 = P and V2 > V1 (expansion).
- PV diagram representation: A horizontal line at P from V1 to V2.
Calculating boundary work for this process:
- Start from the general definition: $Wb = \int{V1}^{V2} P \, dV$
- Because P is constant, pull P out of the integral: $Wb = P \int{V1}^{V2} dV = P (V2 - V1)$
- Sign convention check: Volume increases (expansion), so dV > 0, P > 0; thus W_b > 0, meaning energy is transferred from the system to the surroundings (consistent with the notion of expansion work leaving the system).
Conceptual reminder: The area of the rectangle on the PV diagram represents the boundary work for this simple isobaric process.

Practical implications and connections to real-world engineering

The material connects to real-world devices: rockets, engines, turbines, pumps, nozzles.
In real design work, you often deal with both closed and open systems, depending on whether you’re analyzing a storage tank (closed) or a nozzle/turbine (open).
The specific-property framework (u, v, h) is essential for reading thermo tables and performing energy, enthalpy, and volume balances for different substances.
The sign conventions and energy balance equations underpin thermal system design, performance analysis, and control volume analyses (mass and energy conservation).
Emphasis on understanding the difference between state properties (instantaneous) and process properties (path-dependent) when moving from state 1 to state 2.

Notation recap and common equations to memorize

State and process concepts:
- State: defined by properties at an instant in time (P, T, V, etc.).
- Process: path from state 1 to state 2 through changes in the system.
System classifications:
- Closed system (control mass): no mass transfer across the boundary; mass constant.
- Open system (control volume): mass transfer across the boundary; mass flow in/out occurs.
Heat and work definitions and signs:
- Heat transfer into the system: q > 0
- Work transfer out of the system: w > 0 (boundary work; energy leaving via boundary)
- Energy balance for a closed system: $\Delta U = q - w$
Mass flow rate and energy rates for open systems:
- Mass flow rate: $\dot{m} = \frac{dm}{dt}$
- Heat transfer rate: $\dot{Q}$ (e.g., W or BTU/hr)
- Work rate: $\dot{W}$ (e.g., W or HP)
Specific properties:
- Specific volume: $v = \frac{V}{m}$
- Specific internal energy: $u = \frac{U}{m}$
- Specific enthalpy: $h = \frac{H}{m}$
- Total property recovery: $A = m a$ where a is the corresponding specific property
Boundary work and PV diagrams:
- Boundary work: $Wb = \int{V1}^{V2} P \, dV$
- Isobaric process: $Wb = P (V2 - V_1)$
- Work is the area under the curve on a PV diagram; PV diagrams are a fundamental tool for visualizing energy transfer due to volume changes.

Additional context and references mentioned in the lecture

The instructor referenced Scherzer’s lessons 1–3 and Degani's videos as prerequisites for this material.
The content connects to broader thermodynamics topics such as boundary work, control volume analysis, and the role of state vs. process in energy balances.
Real-world relevance highlighted via SpaceX example to emphasize rapid iteration, system-level thinking, and consideration of large-scale energy systems in engineering contexts.

Quick worked takeaway: a compact example you can reuse

For a closed system undergoing an isobaric expansion from volume $V1$ to $V2$ at constant pressure $P$, the boundary work is:
- $Wb = P (V2 - V_1)$
If you know the mass and a specific property, you can recover the total property:
- $A = m a$
For a process path from state 1 to state 2, the energy balance (for a closed system) is:
- $\Delta U = Q - W$
- If considering boundary work only (no other work forms), then $W = Wb = \int{V1}^{V2} P \, dV$
In open systems, be mindful of mass flow terms; the same PV relationships apply to boundary work, but you must also account for energy carried by mass flow across the boundary (enthalpy flux, kinetic energy, potential energy, etc., depending on the level of modeling).