Thermodynamic Properties and Energy-Balance Experiments

Introduction and Context

Thermal engineering laboratories connect thermodynamic theory with measurable behavior. In a laboratory, pressure, temperature, mass, volume, electrical power, and time are observed; thermodynamic properties and energy transfers are then inferred from these observations. The central task is not merely to obtain a numerical answer, but to establish a defensible chain from measurement to model, from model to balance equation, and from balance equation to engineering conclusion.

This chapter develops experimental methods for determining thermophysical properties and applying mass, energy, entropy, and exergy balances to closed and steady-flow systems. Typical equipment includes pressure–temperature instruments, calorimeters, electrical heaters, flow meters, thermocouples, data-acquisition systems, and property databases or software.

A thermodynamic property describes the state of a system. Examples include pressure, temperature, specific volume, density, internal energy, enthalpy, entropy, and specific heats. A balance expresses conservation or degradation: mass is conserved, energy is conserved, entropy is generated, and exergy is destroyed by irreversibility.

Learning Objectives

After completing this chapter, a student should be able to:

1. Distinguish intensive and extensive properties and identify state variables.
2. Use pressure–temperature data and property tables or databases to estimate thermodynamic properties.
3. Calibrate and operate a simple calorimeter and determine specific heat or latent heat.
4. Formulate mass and energy balances for closed systems and steady-flow control volumes.
5. Apply entropy and exergy balances to assess irreversibility and experimental performance.
6. Quantify uncertainty, identify systematic errors, and report results with appropriate units and significant figures.
7. Judge whether experimental results are physically consistent with the first and second laws.

1. Thermodynamic State and Properties

A system is a specified quantity of matter or a region in space selected for analysis. A closed system permits energy transfer but not mass transfer across its boundary. A control volume is an open system through which mass may flow.

A property is a measurable or calculable characteristic of a system. Properties are classified as follows:

For a homogeneous substance, a simple compressible state is commonly specified by two independent intensive properties. Thus, pressure and temperature can define the state of a single-phase substance, but not generally a saturated liquid–vapour mixture unless quality or another independent property is also known.

The specific volume and density are related by

where v is specific volume in m³/kg, V is volume in m³, m is mass in kg, and ρ is density in kg/m³. This relation applies to a uniform substance for which the measured volume and mass represent the same system.

The total energy of a system is conventionally written as

where U is internal energy, KE = mC²/2 is macroscopic kinetic energy, PE = mgz is gravitational potential energy, C is centre-of-mass velocity, g is gravitational acceleration, and z is elevation. The symbol U is not the same as μ; internal energy is an extensive property, while specific internal energy is u = U/m.

Pressure and temperature measurement

Absolute pressure is measured relative to a perfect vacuum; gauge pressure is measured relative to local atmospheric pressure:

The equation applies when the gauge instrument is correctly zeroed and atmospheric pressure is known. Pressure must be expressed in absolute units in equations involving thermodynamic state relations, such as the ideal-gas equation.

For an ideal gas,

where P is absolute pressure in Pa, v is specific volume in m³/kg, R is the specific gas constant in J/(kg·K), and T is absolute temperature in K. The ideal-gas model is appropriate when intermolecular effects are sufficiently small, commonly at low pressure and sufficiently high temperature relative to the substance’s saturation conditions.

Temperature differences may be expressed in kelvins or degrees Celsius because 1 K = 1 °C in magnitude. Absolute temperature in thermodynamic equations must be in kelvins:

Property databases and interpolation

Measured pressure and temperature are frequently entered into steam tables, refrigerant databases, equations of state, or software libraries. A sound procedure is:

1. Identify the working fluid and its purity.
2. Convert all readings to consistent units.
3. Determine the phase region using saturation data or a phase diagram.
4. Select the appropriate table: compressed liquid, saturated mixture, superheated vapour, or supercritical fluid.
5. Interpolate only between nearby tabulated values.
6. Record the source, reference state, version, and estimated uncertainty.

For linear interpolation of a property y between x₁ and x₂,

This formula assumes that the property varies approximately linearly over the interval. Interpolation should not cross a phase boundary without separate treatment.

2. Calorimetry and Thermophysical Properties

A calorimeter measures heat transfer indirectly through a known temperature change or phase change. A practical calorimeter includes a vessel, insulation, a stirrer, a thermometer or thermocouple, and often an electrical heater. Perfect insulation is impossible, so calibration is essential.

Sensible heating and specific heat

For a substance undergoing a temperature change without phase change, the heat capacity is defined by

where C is heat capacity in J/K and δQ is an infinitesimal heat transfer. If the specific heat is approximately constant over the measured range,

where Q is energy transferred as heat in J, m is mass in kg, c_p is specific heat at constant pressure in J/(kg·K), and ∆T = T₂ − T₁. The equation is appropriate when changes in kinetic and potential energy are negligible, pressure effects are represented by c_p, and c_p does not vary substantially over the temperature interval.

For a calorimeter containing a sample and calorimeter body, an energy balance may be written as

where m_s and c_s are sample mass and specific heat, C_cal is calorimeter heat capacity, and Q_loss is heat lost to the surroundings. If the calorimeter is calibrated and losses are corrected, the sample specific heat is

The units are J/(kg·K) when Q_in, C_cal∆T, and Q_loss are in joules, mass is in kilograms, and temperature difference is in kelvins.

Electrical calibration

When an electrical heater supplies energy, the input energy over a measured interval is

where V is voltage in volts, I is current in amperes, and t is time in seconds. If voltage and current are effectively constant,

Electrical energy is converted to heat with an efficiency η_e:

The efficiency includes heater and wiring losses and must be determined or justified experimentally. Assuming η_e = 1 without calibration generally overestimates the heat received by the specimen.

Latent heat

During a phase change at approximately constant pressure and saturation temperature, the heat transfer is

where L is latent heat in J/kg. For vaporization, L = h_g − h_f, where h_g and h_f are saturated-vapour and saturated-liquid specific enthalpies at the same pressure. The relation applies when sensible heating outside the phase-change interval and heat losses have been accounted for.

Worked Example 1: Specific heat from a calibrated calorimeter

An electrically heated calorimeter receives 18.0 kJ. Its calibrated heat capacity is 0.42 kJ/K. A 0.250 kg metal specimen causes the measured temperature to rise by 32.0 K. Heat loss during the test is estimated as 1.20 kJ. Determine the specimen specific heat.

The energy balance is

The calorimeter energy is 13.44 kJ. Therefore,

Thus, the estimated specific heat is 420 J/(kg·K). The result should be compared with a reference value only after considering specimen composition, temperature dependence, calibration uncertainty, and heat-loss estimation.

3. Mass and Energy Balances

General balance structure

A balance begins with a clearly defined system boundary and time interval. The general mass balance is

where m_CV is mass inside a control volume and ṁ is mass flow rate in kg/s. For a closed system, there is no mass crossing the boundary and total mass remains constant.

The first-law energy balance for a control volume is

Here, Q̇ is heat-transfer rate into the control volume, Ẇ is work-transfer rate out of the control volume, h is specific enthalpy, C is flow speed, g is gravitational acceleration, and z is elevation. The equation assumes one-dimensional flow at each inlet and outlet and uses the convention that heat input is positive and work output is positive.

For steady operation, dE_CV/dt = 0. If changes in kinetic and potential energy are negligible and there is one inlet and one outlet,

For an adiabatic throttling valve with no shaft work,

For a steady-flow heater with negligible pressure work beyond the enthalpy definition and no shaft work,

Closed-system energy balance

For a closed system, the first law is

or, if changes in kinetic and potential energy are negligible,

where W_b is boundary work done by the system and W_other represents other work modes. For a quasi-equilibrium piston–cylinder process,

This integral applies when system pressure is well defined throughout the process. It cannot be replaced by W_b = P∆V unless pressure is constant or a justified average-pressure approximation is used.

Measurement of flow energy

A flow experiment usually requires mass flow rate. A timed collection method gives

where ∆m is collected mass and ∆t is collection time. Repeated measurements are recommended because short collection intervals magnify stopwatch and balance errors.

4. Entropy and Exergy Balances

The second law introduces entropy generation, which measures irreversibility. For a control volume,

where S_CV is entropy inside the control volume, Q̇_j is heat transfer at boundary temperature T_j in kelvins, s is specific entropy in J/(kg·K), and Ṡ_gen is entropy-generation rate. The second law requires

A negative calculated value normally indicates measurement uncertainty, an incorrect sign convention, an inappropriate property value, or an invalid steady-state assumption.

At steady state with one inlet, one outlet, and heat transfer at a uniform boundary temperature,

where T_b is the absolute boundary temperature. For an adiabatic steady-flow device,

Exergy balance

Exergy is the maximum useful work relative to a specified environment, called the dead state. It is not conserved because irreversibility destroys the opportunity to do work. The exergy destruction rate is related to entropy generation by the Gouy–Stodola relation:

where T₀ is ambient absolute temperature and Ẋ_destroyed is in watts when T₀ is in kelvins and Ṡ_gen is in W/K.

For a steady-flow device, the physical-flow exergy per unit mass may be expressed as

where subscript 0 denotes dead-state properties. Chemical exergy is omitted in this expression; it must be included for reacting or composition-changing systems. Exergy analysis is especially useful for distinguishing energy quantity from energy quality: heat supplied at a high temperature has greater work potential than the same heat supplied near ambient temperature.

5. Experimental Methodology and Uncertainty

A reliable laboratory result requires more than a correct equation. The apparatus should be inspected for leaks, electrical hazards, loose thermocouple connections, damaged insulation, and pressure exceeding instrument ratings. The experiment should be run until the stated steady-state criterion is reached, not merely until readings appear convenient.

Calibration and traceability

Calibration compares an instrument with a reference of known or certified value. A calibration curve may be represented by

where x is the instrument indication, y is the corrected value, and a and b are calibration constants. Calibration should cover the range used in the experiment. A single-point check cannot establish linearity or hysteresis.

Uncertainty propagation

For a result y = f(x₁, x₂, …, x_n), an approximate combined standard uncertainty for independent variables is

where u_xi is the standard uncertainty of input x_i. This first-order expression assumes small uncertainties and negligible correlation. If inputs are correlated, covariance terms must be added.

For multiplication or division, relative uncertainty is often estimated by

for y = ab or y = a/b, provided a and b are independent and nonzero. Report the result as, for example, c_p = 420 ± 15 J/(kg·K), with the uncertainty basis and coverage statement identified.

Common systematic errors include heat leakage, sensor calibration drift, thermal stratification, pressure transducer zero error, unaccounted kinetic-energy changes, and use of gauge rather than absolute pressure. Random errors arise from reading fluctuations, timing variation, and repeatability limits. Repeating a measurement reduces random uncertainty but does not remove systematic bias.

6. Illustrative Case: Steady-Flow Heater Evaluation

Background

A laboratory team evaluates a small electric flow heater that warms water at a nearly steady flow rate. The objective is to determine thermal efficiency and check the entropy balance.

Problem

Water enters at 20.0 °C and leaves at 60.0 °C. The measured mass flow rate is 0.020 kg/s. Electrical input is 4.00 kW. Pressure changes and kinetic-energy changes are negligible. The specific heat of liquid water is approximated as 4.18 kJ/(kg·K). Heat loss to the room is estimated as 0.35 kW.

Analysis and solution

The useful rate of energy gained by the water is

Thus,

The measured balance is consistent because useful heating plus estimated heat loss is

The remaining 0.306 kW may represent electrical measurement uncertainty, unmodelled storage, wiring loss, or an inaccurate heat-loss estimate. Taking electrical input as the supplied energy, the water-heating efficiency is

or 83.6%.

For an approximate entropy-generation calculation, treat the heater as adiabatic with respect to the water control volume and use constant c_p. For an incompressible liquid,

with temperatures in kelvins. Therefore,

The entropy increase rate of the water is

If the heater transfers energy internally at an effective boundary temperature, a complete entropy balance would include that heat-transfer term. The positive entropy increase is physically plausible because finite-temperature heat transfer and electrical resistance are irreversible.

Outcome and learning points

The heater transfers approximately 3.34 kW to the water and achieves an experimental water-heating efficiency of 83.6%. The difference between electrical input and useful water heating must not automatically be labelled “heat loss”; it may include instrument error and energy stored in the apparatus. The case demonstrates why energy balances and uncertainty analysis must be interpreted together.

7. Best Practices, Limitations, and Contemporary Tools

Best practice is to define the system boundary before taking data, use absolute pressure and kelvin temperature where required, record raw readings rather than only calculated values, repeat measurements, and preserve calibration information. Property software is valuable, but its output is not automatically correct. Users must verify fluid identity, phase region, reference state, units, and interpolation behavior.

Modern laboratories increasingly use digital thermocouples, automated data acquisition, internet-connected sensors, and real-time plotting. Artificial-intelligence tools can assist with anomaly detection, regression, and uncertainty screening, but they cannot replace physical validation. A model that predicts a negative entropy generation or violates a known conservation law must be investigated rather than accepted because it fits measured data.

Safety is part of professional competence. Pressurized vessels require rated components, relief protection, inspection, and controlled depressurization. Electrical heaters require grounding, over-current protection, dry working conditions, and isolation before adjustment. Hot surfaces, steam, refrigerants, and rotating equipment require suitable personal protective equipment and operating procedures. Laboratory records should be honest: data should not be deleted merely because they disagree with theory.

Chapter Summary

Thermodynamic experiments determine properties by combining measurement with a physically appropriate model. Pressure and temperature identify states only when phase conditions and independence requirements are satisfied. Calorimeters determine sensible or latent heat effects, but calibration and heat-loss correction are essential. The first law provides mass and energy balances for closed systems and control volumes. The second law requires non-negative entropy generation, while exergy analysis measures the work potential destroyed by irreversibility. Accurate laboratory work depends on correct boundaries, consistent units, calibrated instruments, repeatability, uncertainty analysis, and critical comparison with property data. Energy-balance results should always be tested for physical plausibility and conservation-law consistency.

Key Terms

Thermodynamic property: A measurable characteristic of a system defined by its state.

Intensive property: A property independent of the amount of substance.

Extensive property: A property that depends on system size.

Specific property: An extensive property per unit mass.

State: The condition of a system described by its properties.

Closed system: A system across whose boundary mass does not pass.

Control volume: A selected region in space through which mass may flow.

Calorimeter: An apparatus used to measure heat transfer indirectly.

Specific heat: Heat capacity per unit mass under a specified condition.

Latent heat: Energy per unit mass associated with a phase change.

Enthalpy: The property h = u + Pv for a simple compressible substance.

Mass flow rate: Mass crossing a boundary per unit time.

Entropy: A state property used to quantify energy dispersal and second-law behavior.

Entropy generation: Entropy produced by irreversibility.

Exergy: Maximum theoretical useful work relative to an environment.

Dead state: The reference environmental state used in exergy analysis.

Calibration: Comparison of an instrument with a known reference.

Uncertainty: Quantified doubt associated with a measurement or result.

Systematic error: A repeatable bias that is not removed by averaging.

Steady state: A condition in which control-volume properties do not change with time.

Review Questions

Short Answer Questions

1. Distinguish an intensive property from an extensive property and give one example of each.
2. Why must absolute pressure be used in the ideal-gas equation?
3. State the assumptions required for using Q = mc_p∆T.
4. What physical meaning does entropy generation have?
5. Why is calorimeter calibration necessary even when the calorimeter is insulated?

Long Answer Questions

1. Explain how pressure–temperature measurements are converted into thermodynamic properties using a property database. Include phase identification and interpolation.
2. Derive the simplified steady-flow energy equation for a single-inlet, single-outlet device with negligible kinetic and potential energy changes.
3. Describe a complete electrical calorimetry experiment for determining the specific heat of a solid specimen.
4. Compare energy analysis and exergy analysis for a steady-flow heater.
5. Discuss random and systematic errors in a flow-heating experiment and propose methods for reducing each type.

Analytical / Application Questions

1. A 0.300 kg specimen receives 25.0 kJ, the calorimeter heat capacity is 0.50 kJ/K, the temperature rise is 28.0 K, and estimated heat loss is 1.5 kJ. Determine the specimen specific heat and identify two additional measurements needed to establish uncertainty.
2. Water flows steadily at 0.015 kg/s through a heater, entering at 25 °C and leaving at 75 °C. Using c_p = 4.18 kJ/(kg·K), calculate the useful heating rate. If electrical input is 3.50 kW, calculate the water-heating efficiency.
3. An experimental steady-flow device produces a negative entropy-generation rate. Develop a diagnostic sequence that checks units, pressure reference, property selection, heat-transfer sign, steady-state assumptions, and measurement uncertainty before rejecting the data.

Multiple Choice Questions with Answers

Question 1

Which property is intensive?

A. Mass

B. Total volume

C. Temperature

D. Total internal energy

Correct Answer: C. Temperature

Question 2

In the ideal-gas relation Pv = RT, pressure must generally be expressed as

A. Gauge pressure

B. Absolute pressure

C. Differential pressure only

D. Vacuum pressure only

Correct Answer: B. Absolute pressure

Question 3

A calorimeter receives 10 kJ and its contents have an effective heat capacity of 0.50 kJ/K. Neglecting losses, the temperature rise is

A. 5 K

B. 10 K

C. 20 K

D. 50 K

Correct Answer: C. 20 K

Question 4

For a steady-flow heater with negligible kinetic and potential energy changes and no shaft work, the useful heating rate is approximately

A. ṁ(h₂ − h₁)

B. ṁ(h₁ + h₂)

C. ṁ/(h₂ − h₁)

D. (h₂ − h₁)/ṁ

Correct Answer: A. ṁ(h₂ − h₁)

Question 5

The second law requires entropy generation to be

A. Always negative

B. Always zero

C. Non-negative

D. Equal to heat input

Correct Answer: C. Non-negative

Question 6

The Gouy–Stodola relation is

A. Ẋ_destroyed = T₀Ṡ_gen

B. Ẋ_destroyed = Ṡ_gen/T₀

C. Ẋ_destroyed = T₀/Ṡ_gen

D. Ẋ_destroyed = Q̇/T₀

Correct Answer: A. Ẋ_destroyed = T₀Ṡ_gen

Question 7

A flow experiment collects 3.0 kg in 120 s. The mass flow rate is

A. 0.0025 kg/s

B. 0.025 kg/s

C. 0.25 kg/s

D. 2.5 kg/s

Correct Answer: A. 0.0025 kg/s

Question 8

Which is a systematic error?

A. Random reading fluctuation

B. A consistently miscalibrated pressure sensor

C. Variation in stopwatch reaction time

D. Electrical noise that changes unpredictably

Correct Answer: B. A consistently miscalibrated pressure sensor

Question 9

For vaporization at a specified saturation pressure, latent heat is represented by

A. h_f + h_g

B. h_f − h_g

C. h_g − h_f

D. h_g/h_f

Correct Answer: C. h_g − h_f

Question 10

A negative experimental entropy-generation rate most appropriately suggests

A. The second law has failed

B. The data or model should be checked for errors and uncertainty

C. The device is perfectly reversible

D. Energy conservation is unnecessary

Correct Answer: B. The data or model should be checked for errors and uncertainty