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1.4 · Interpret schematic P–v and T–s process diagrams

Learn to interpret schematic p–v and t–s process diagrams through clear examples and targeted practice.

University of Alberta MEC E 340: Applied Thermodynamics

Cycle Analysis Foundations

MEC E 340 study topic 1.4

A P–v or T–s diagram is a way to represent thermodynamic states and the processes connecting them. For this topic, consider a fixed mass of working fluid moving through numbered states in a closed cycle, or a specified fluid system undergoing an open state-to-state process. The plots are schematic unless exact coordinates or property data are supplied: their spacing and shape do not provide numerical values by themselves. Use pressure vertically and specific volume horizontally on a P–v plot; use temperature vertically and specific entropy horizontally on a T–s plot. The goal is to read what the diagram supports without assuming more than it shows.

What you will learn

Identify what each axis and plotted state represents on P–v and T–s diagrams.
Use the direction of a process or cycle to describe how pressure, specific volume, temperature, and entropy change.
Interpret the area enclosed by a P–v cycle and the area under a T–s process only when the required assumptions apply.
Separate conclusions supported by a schematic diagram from quantities that require property data.

1. Read the axes, states, and direction

Start by deciding what system the diagram represents. In a closed cycle, the working fluid returns to its initial state after passing through numbered states such as 1, 2, and 3. For a state-to-state process, the system moves from an initial state to a final state and does not necessarily return. A state is described by properties; a process is the change between states.

On a P–v diagram, pressure,

P

, is vertical and specific volume,

v

, is horizontal. Moving upward means pressure increases; moving right means specific volume increases. On a T–s diagram, temperature,

T

, is vertical and specific entropy,

s

, is horizontal. Moving upward means temperature increases; moving right means entropy increases. Follow arrows or stated process order to determine the direction. A line without arrows does not establish which way the process proceeds.

The path shape may suggest a process constraint only when the problem identifies it. For example, a horizontal segment on a P–v plot indicates constant pressure in that representation. Do not identify an unlabeled curve as a particular process just because it resembles one. A schematic plot may not have accurate scales, and a drawn line does not supply exact property values.

Read axis labels and units before interpreting a path.
Arrows or stated state order establish direction.
A schematic curve does not, by itself, identify a process model or numerical properties.

2. What areas and cycle direction mean

For a quasi-equilibrium boundary process of a closed system, the area under a P–v path represents boundary work per unit mass, with expansion work positive under the usual convention. A path moving toward larger specific volume gives positive boundary work; compression gives negative boundary work. The numerical work cannot be found from a schematic sketch unless sufficient pressure–volume data or a specified relation is provided.

For a closed cycle, the signed area enclosed on a P–v diagram represents net boundary work per unit mass when the plotted path represents the relevant quasi-equilibrium processes. The direction matters: a clockwise loop commonly indicates positive net work by the system under this convention; a counterclockwise loop indicates negative net work. This interpretation is not a replacement for checking the stated sign convention and process information.

On a T–s diagram, the area under a process curve represents heat transfer per unit mass only for an internally reversible process. For an internally reversible cycle, the signed area enclosed corresponds to net heat transfer per unit mass. Do not infer heat transfer from T–s area if reversibility is not stated or established. Neither plot alone gives efficiency, COP, or exact heat and work values without the needed property data and energy balances.

w_b=\int P\,dv

P–v area relates to boundary work under the stated process assumptions.
T–s area relates to heat transfer only for internally reversible processes.
Cycle direction and sign convention must be interpreted together.

3. A disciplined interpretation

Before interpreting a plot, list the working fluid, the system boundary, numbered states, process direction, and any assumptions supplied. For a closed system, mass is fixed; for a control volume, mass may cross the boundary, so a cycle sketch should not be mistaken for a device-flow diagram. This lesson focuses on reading process plots, not calculating device performance.

Use the first law only when the question asks for energy transfer or work and enough information is available. For a cycle, the fluid returns to its initial state, so its net change in stored energy over a complete cycle is zero. For an individual process, property differences or other supplied data are needed to evaluate energy changes. The plot can show qualitative trends, but exact values must come from given data or a named property source.

A reliable interpretation separates three levels: what the axes directly show, what follows from a stated physical assumption, and what remains unknown. For example, a path moving right on P–v shows increasing specific volume; calling it expansion work requires the closed-system boundary-work interpretation; assigning a work value requires numerical pressure–volume information.

\Delta E_{\mathrm{cycle}}=0

Distinguish closed-system cycle plots from control-volume device diagrams.
Use balances to support calculations, not to invent missing properties.
State clearly which conclusions are qualitative and which are calculated.

4. Common visual traps

A steep line is not necessarily a large energy transfer: the axes may use different scales, and the plot may be schematic. A horizontal or vertical segment has a direct meaning only with respect to its labeled axes. It does not automatically imply a familiar named process beyond the property held constant on that plot.

The locations of states on a diagram do not establish phase unless phase information or a suitable property diagram and data are supplied. In particular, do not add a saturation boundary or assign liquid, vapour, or mixture states from an unlabeled schematic. Likewise, do not infer an exact temperature, pressure, entropy, or volume from the distance between points.

When asked to compare two processes, identify the relevant interval and compare the plotted coordinate or path area only if the required conditions hold. If the scale, process assumption, or property data are missing, say what can be concluded and name what additional information would be needed.

Sketch scale and curve shape may not be quantitative.
Do not assign phase or process type without supporting information.
A correct answer can state that the diagram alone is insufficient.

Worked example

Reading a schematic closed cycle

A closed system completes the three-state loop 1→2→3→1 shown schematically on a P–v plot. The path moves from state 1 to 2 toward larger specific volume, from 2 to 3 toward smaller pressure, and returns from 3 to 1. The supplied plot shows a clockwise loop. State what can be inferred about the net boundary work, and whether its numerical value can be found.

Schematic P–v cycle

Closed-system working fluid; schematic and not to scale. The stated clockwise loop is used only to indicate the sign of net boundary work.

Identify the system and axes
The system is the fixed mass completing a cycle. The horizontal coordinate is specific volume and the vertical coordinate is pressure. The stated state order and clockwise direction define the loop.
Interpret the signed area
For the usual positive-work-by-the-system convention, the clockwise enclosed P–v area indicates positive net boundary work by the system over the cycle. The sketch provides no scale or numerical pressure–volume data, so it does not determine the work magnitude.
$W_{b,\mathrm{net}}>0$

Answer: The net boundary work is positive under the usual convention, but its numerical value cannot be determined from the schematic alone.

Check: The conclusion uses loop direction for the sign and does not mistake schematic area for a measured numerical area.

Worked example

Reading property changes on an open path

A closed-system working fluid moves from state 1 to state 2 along the supplied schematic P–v path. State 2 is to the right and above state 1. What property changes are established by the plot, and what remains unknown?

Schematic P–v state change

Closed-system working fluid from state 1 to state 2; schematic and not to scale. No phase or process constraint is supplied.

Compare the state coordinates
On P–v axes, moving right means larger specific volume and moving upward means greater pressure. The plot therefore establishes increases in both properties from state 1 to state 2.
$v_2>v_1,\qquad P_2>P_1$
Limit the conclusion
The diagram does not establish the temperature change, phase, or numerical property differences. Those conclusions need additional property information or a specified model. The path direction also does not alone provide a numerical boundary-work value.

Answer: Specific volume and pressure increase. Temperature, phase, and numerical changes are not determined by the given schematic.

Check: The answer uses only the two labeled coordinates and does not infer an unsupported process type.

Worked example

Using a T–s plot cautiously

A working fluid undergoes a process from state 1 to state 2 on a T–s diagram. The path is drawn horizontally to the right, and the problem explicitly states that the process is internally reversible. State what the diagram establishes and whether the heat transfer can be calculated from the sketch.

Schematic T–s process

Working fluid from state 1 to state 2; internally reversible as stated. Schematic and not to scale.

Read the coordinates
Temperature is vertical and entropy is horizontal. A horizontal path indicates constant temperature on this plot, while moving right indicates increasing entropy.
$T_2=T_1,\qquad s_2>s_1$
Apply the reversibility condition
Because internal reversibility is explicitly given, heat transfer per unit mass can be related to the area under the T–s path. The sketch has no numerical temperature or entropy scale, so the heat amount cannot be calculated from it.
$q_{1\to2}=\int_1^2 T\,ds$

Answer: Temperature is constant and entropy increases. Reversibility allows the T–s area interpretation, but the schematic supplies no numerical area, so heat transfer cannot be evaluated.

Check: The heat-area interpretation is used only because internal reversibility is stated.

Common mistakes and how to avoid them

Treating the apparent curve shape or spacing as exact data.

Correction: Read qualitative trends from schematic coordinates; calculate values only from supplied scales, properties, or a stated relation.

Using T–s area to calculate heat for any process.

Correction: Use that area interpretation only when the process is internally reversible.

Assigning phase or a named process from an unlabeled path.

Correction: Require supporting phase information or an explicitly stated process constraint.

Lesson summary

P–v plots have pressure vertical and specific volume horizontal; T–s plots have temperature vertical and entropy horizontal.
Follow state numbers and arrows to determine process direction and property trends.
P–v area represents boundary work under the appropriate closed-system process interpretation.
T–s area represents heat transfer only for internally reversible processes.
Schematic diagrams support qualitative conclusions, not invented coordinates, properties, or performance values.

Check your understanding

Question 1

On a P–v plot, a directed path moves left. What property trend is directly shown?

Specific volume decreases.
Pressure decreases.
Entropy increases.
Temperature is constant.

Show answer and explanation

Specific volume decreases.

Specific volume is the horizontal coordinate, so moving left means it decreases.

Question 2

When may the area under a T–s process path be interpreted as heat transfer per unit mass?

Whenever the working fluid completes a cycle.
Only when internal reversibility applies to the process.
Whenever the path is drawn as a straight line.
Only when pressure is constant.

Show answer and explanation

Only when internal reversibility applies to the process.

The T–s heat-area interpretation requires an internally reversible process.

Question 3

A schematic P–v cycle is clockwise. Under the usual positive-work-by-the-system convention, what can be concluded?

Net boundary work is positive, but its magnitude needs quantitative data.
Net boundary work is negative and its magnitude is shown by the sketch.
Net heat transfer must be zero.
The working fluid must be vapour.

Show answer and explanation

Net boundary work is positive, but its magnitude needs quantitative data.

Clockwise enclosed P–v area indicates positive net boundary work under the stated convention; a schematic has no dependable numerical scale.

Key terms

State: A condition of the working fluid described by its thermodynamic properties.
Process: A change of the system from one thermodynamic state to another.
Specific volume: Volume per unit mass, denoted by $v$ .
Specific entropy: Entropy per unit mass, denoted by $s$ .
Schematic diagram: A qualitative plot intended to show relationships or process order, not necessarily exact scale or property values.

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