3. First-Law Analysis of Closed and Open Systems

Introduction and Context

The first law of thermodynamics is an energy-accounting principle. It states that energy cannot be created or destroyed, although it may cross a system boundary, be stored within a system, or change from one form to another. In engineering analysis, the law provides the basis for determining heat transfer, work interaction, power output, fuel or fluid requirements, and changes in stored energy.

The distinction between closed and open systems is central. A closed system contains a fixed quantity of matter; energy may cross its boundary, but mass does not. An open system, or control volume, permits both mass and energy to cross its boundary. Boilers, turbines, compressors, pumps, nozzles, heat exchangers, mixing chambers, and throttling valves are therefore analyzed as open systems.

The first-law analysis is not a substitute for a second-law analysis. It determines whether an energy balance is satisfied, but it does not establish the direction of a process, its reversibility, or the maximum possible performance. Nevertheless, first-law analysis is the indispensable first step in the design and evaluation of thermal and fluid machinery.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Formulate energy balances for closed systems and control volumes.

2. Distinguish stored, transferred, and flow energy.

3. Apply the general mass and energy balance to steady and transient processes.

4. Analyze turbines, compressors, pumps, nozzles, throttles, mixing chambers, heat exchangers, and storage devices.

5. State and justify the assumptions used in engineering simplifications.

6. Evaluate component performance using quantities such as work, heat transfer, efficiency, and pressure drop.

7. Recognize common sign, unit, reference-state, and property-data errors.

8. Use computational tools responsibly while preserving physical interpretation of results.

Energy Accounting: System, Boundary, and State

A thermodynamic system is a specified quantity of matter or a region in space selected for analysis. Its surroundings are everything external to it, and the boundary separates the two. The boundary may be fixed, moving, real, or imaginary. A control volume is an open-system boundary selected around a device or process.

The total energy of a system is commonly written as

E = U + KE + PE

where U is internal energy, KE is kinetic energy, and PE is potential energy. For a system containing mass m,

E = U + mV^2/2 + mgz.

On a mass-specific basis,

e = u + V^2/2 + gz.

Internal energy is associated with molecular translation, rotation, vibration, chemical bonding, and other microscopic modes. Kinetic energy depends on macroscopic velocity, while potential energy depends on elevation relative to a selected datum. The datum for potential energy is arbitrary, provided it is used consistently.

Energy transfer across a boundary occurs as heat or work. Heat is energy transfer caused solely by a temperature difference. Work is energy transfer associated with a generalized force acting through a displacement or with other organized interactions. Electrical work, shaft work, boundary work, and flow work are important in engineering systems.

A useful convention is to take heat transfer to the system as positive and work done by the system as positive. With this convention, the first law for a closed system is

Q - W = Delta E

or, for a process between states 1 and 2,

Q_1-2 - W_1-2 = (U_2-U_1) + (KE_2-KE_1) + (PE_2-PE_1).

Heat and work are path functions, not properties. Their values depend on how the process occurs. Internal energy, enthalpy, pressure, temperature, and specific volume are properties determined by the state.

Closed-System Analysis

For a closed system, no mass crosses the boundary. The general energy balance is

Energy entering - Energy leaving = Increase in stored energy.

When heat and work are the only energy transfers, this becomes

Q_1-2 - W_1-2 = Delta U + Delta KE + Delta PE.

For a stationary closed system, changes in kinetic and potential energy are negligible, giving

Q_1-2 - W_1-2 = Delta U.

For a quasi-equilibrium compression or expansion of a simple compressible substance, boundary work is

W_b = integral from 1 to 2 of p dV.

If pressure is constant, W_b = p(V_2 - V_1). For an ideal gas undergoing a constant-pressure process, the first law may be written using Delta U = m c_v Delta T. If only boundary work is present,

Q = Delta U + W_b.

At constant pressure this leads to Q = Delta H, where H = U + pV is enthalpy. This result is especially useful for closed-system heating processes, but it must not be generalized to every constant-pressure process without checking the work modes and system definition.

Internal Energy and Enthalpy

Internal energy is the appropriate stored-energy property for a closed stationary system. Enthalpy is defined by

H = U + pV,

and per unit mass,

h = u + pv.

Enthalpy is not a new form of energy. It combines internal energy with the pV term, which becomes convenient when matter crosses a control surface. For ideal gases, internal energy and enthalpy are primarily functions of temperature. Thus,

Delta u = integral c_v(T)dT, Delta h = integral c_p(T)dT.

When specific heats are treated as constant, Delta u = c_v(T_2-T_1) and Delta h = c_p(T_2-T_1), with c_p - c_v = R.

The Control-Volume Energy Balance

An open system must account for energy carried by mass flow. The general rate form of the first law for a control volume is

dE_cv/dt = Q_dot_cv - W_dot_cv + sum_in m_dot (h + V^2/2 + gz) - sum_out m_dot (h + V^2/2 + gz).

Here W_dot_cv represents work other than the flow work included in enthalpy. In the common convention, heat entering and work leaving are positive in the displayed form. A consistent alternative sign convention is equally valid.

The corresponding mass balance is

dm_cv/dt = sum_in m_dot - sum_out m_dot.

For a steady-flow control volume, properties within the control volume do not change with time, so dm_cv/dt = 0 and dE_cv/dt = 0. The steady-flow energy equation becomes

Q_dot - W_dot = sum_out m_dot(h + V^2/2 + gz) - sum_in m_dot(h + V^2/2 + gz).

For one inlet and one outlet with mass flow rate m_dot,

q - w = (h_2-h_1) + (V_2^2-V_1^2)/2 + g(z_2-z_1),

where q and w are energy transfers per unit mass. Alternatively,

w = q + h_1-h_2 + (V_1^2-V_2^2)/2 + g(z_1-z_2).

These equations should be written before assumptions are imposed. The assumptions—steady state, adiabatic operation, negligible changes in velocity or elevation, one-dimensional flow, or negligible leakage—must be justified by the device and operating conditions.

Flow Energy and the Role of Enthalpy

A fluid crossing a control surface must push aside fluid downstream. The associated flow work per unit mass is pv. Combining internal energy and flow work gives enthalpy, u + pv. Consequently, the energy carried by flowing matter is

h + V^2/2 + gz.

This expression explains why enthalpy, rather than internal energy alone, appears in control-volume balances. It also clarifies why enthalpy differences are frequently used to calculate turbine work, compressor work, and heat transfer in steady-flow devices.

The distinction between total and specific quantities is essential. Total enthalpy is H, while specific enthalpy is h. A rate such as m_dot h has units of power. A quantity such as h has units of energy per unit mass. Mixing these forms produces dimensional errors that may appear numerically plausible.

Device Models and Their Energy Balances

Turbines and Expanders

A turbine converts fluid energy into shaft work. For an adiabatic turbine with one inlet and one outlet and negligible kinetic and potential-energy changes,

w_out = h_1 - h_2.

The actual enthalpy drop is smaller than the ideal isentropic enthalpy drop for the same inlet state and exit pressure. The turbine isentropic efficiency is commonly defined as

eta_t = (h_1-h_2,actual)/(h_1-h_2s).

A value below unity reflects internal irreversibilities such as fluid friction, turbulence, leakage, and non-equilibrium expansion. In a multistage turbine, shaft power is the sum of the work produced by all stages, subject to mechanical losses and heat transfer assumptions.

Compressors and Pumps

Compressors raise the pressure of gases, while pumps raise the pressure of liquids. For an adiabatic compressor with negligible changes in kinetic and potential energy,

w_in = h_2 - h_1.

The isentropic compressor efficiency is

eta_c = (h_2s-h_1)/(h_2,actual-h_1).

For an incompressible liquid pump, the enthalpy relation can be expressed approximately through the pressure rise. Neglecting heat transfer and velocity changes,

w_in approximately equals v(P_2-P_1),

for an idealized reversible pump. Actual pump work is greater because of hydraulic, mechanical, and leakage losses. Pump efficiency may be defined as the ideal fluid power divided by the shaft power supplied.

Nozzles and Diffusers

A nozzle converts enthalpy into kinetic energy, whereas a diffuser reduces velocity and raises static pressure. For an adiabatic nozzle with no shaft work and negligible elevation change,

h_1 + V_1^2/2 = h_2 + V_2^2/2.

If the inlet velocity is small, the enthalpy drop determines the exit velocity:

V_2 = [2(h_1-h_2)]^0.5,

when consistent units are used. A diffuser follows the reverse energy conversion: kinetic energy is transformed into enthalpy and pressure. The actual performance depends on friction, boundary-layer separation, shocks, and area variation.

Throttling Valves

A throttling valve produces a large pressure drop with no significant shaft work. Usually the valve is adiabatic, and changes in kinetic and potential energy are negligible. The steady-flow energy equation then gives

h_1 = h_2.

Thus throttling is approximately isenthalpic, not isentropic. Temperature may rise, fall, or remain nearly unchanged depending on the fluid and state. For an ideal gas, enthalpy depends only on temperature, so an ideal-gas throttling process has approximately constant temperature. Real gases may exhibit a Joule–Thomson temperature change.

Mixing Chambers

In a mixing chamber, two or more streams combine to form one or more outlet streams. At steady state, mass conservation gives

sum m_dot_in = sum m_dot_out.

For an adiabatic chamber with negligible kinetic and potential-energy changes and no shaft work,

sum m_dot_in h_in = sum m_dot_out h_out.

If the outlet is a single stream and the fluid is unchanged, the outlet enthalpy is the mass-weighted average of inlet enthalpies. Pressure, phase equilibrium, and composition restrictions must also be satisfied; an energy balance alone may not determine the complete physical state.

Heat Exchangers

A heat exchanger transfers energy between streams without direct mixing. If the outer control volume is adiabatic and changes in kinetic and potential energy are negligible,

sum m_dot h entering = sum m_dot h leaving.

For two streams,

m_dot_h(h_h,in-h_h,out) = m_dot_c(h_c,out-h_c,in).

The heat capacity rates, phase changes, temperature differences, pressure drops, and fouling condition influence performance. A heat exchanger may be analyzed by an energy balance, an effectiveness–NTU method, or a log-mean-temperature-difference method. The first-law balance determines the heat transfer rate but not by itself the required area or temperature distribution.

Storage Devices and Filling or Emptying Processes

Tanks, receivers, accumulators, and pressurized vessels are transient control volumes. During filling, mass and energy accumulate; during emptying, they decrease. For a rigid, adiabatic tank with one inlet and no outlet,

dU_cv/dt = m_dot_in h_in,

dm_cv/dt = m_dot_in,

when changes in kinetic and potential energy and shaft work are negligible. Integrated over a filling process,

U_2-U_1 = m_in h_in,

where m_in is the total entering mass. The final state must satisfy both the mass balance and an equation of state or property relation.

For a tank discharging through an outlet, the energy equation includes the enthalpy carried away:

dU_cv/dt = -m_dot_out h_out + Q_dot - W_dot.

The exit state is not automatically equal to the average tank state. It depends on mixing, stratification, flow arrangement, and the modeling assumptions used.

Transient Analysis and Integrated Balances

Transient analysis is required when the conditions inside a control volume change appreciably with time. The integral mass balance is

m_2-m_1 = m_in-m_out.

The integral energy balance is

E_2-E_1 = Q-W + sum m_in(h+V^2/2+gz) - sum m_out(h+V^2/2+gz).

For a control volume with a single inlet and no outlet, negligible kinetic and potential energy, and no work,

U_2-U_1 = Q + m_in h_in.

Transient problems often require simultaneous solution of mass conservation, energy conservation, and property relations. If the fluid is a two-phase mixture, quality is useful only when equilibrium conditions apply. Treating a non-equilibrium mixture as a uniform equilibrium state can cause significant error.

A Reliable Procedure for First-Law Analysis

A disciplined solution begins by defining the system or control volume. The analyst should draw all heat, work, and mass interactions and select a sign convention. Next, the general mass and energy balances should be written before deleting terms. The relevant inlet and outlet states are then identified, and thermodynamic properties are obtained from reliable tables, equations of state, or validated software.

The assumptions should be stated explicitly. Typical assumptions include steady state, adiabatic walls, negligible potential-energy change, uniform velocity profiles, negligible leakage, and ideal-gas or incompressible-fluid behavior. Each assumption should be tested against the scale of the neglected term. For example, velocity changes may be negligible in a heat exchanger but dominant in a nozzle.

Finally, the result should be checked dimensionally and physically. A turbine should normally produce positive work, an adiabatic compressor should require work input, and an insulated heat exchanger should conserve total enthalpy flow. Unrealistic temperatures, negative flow rates, or efficiencies greater than unity usually indicate an incorrect sign convention, inconsistent property data, or an invalid assumption.

Applied Case Study: Energy Balance of a Steam-Turbine Stage

Background

Consider an illustrative, steady-flow turbine stage receiving superheated steam at 3 MPa and 500°C and exhausting at 0.1 MPa. The turbine is insulated, and changes in elevation are negligible. The inlet velocity is 40 m/s and the exit velocity is 120 m/s. The mass flow rate is 5 kg/s. Property values are to be obtained from appropriate steam tables or validated software.

Situation and Problem

Determine the specific work output and shaft-power output if the inlet and exit specific enthalpies are h_1 and h_2, respectively. Explain the effect of the velocity change and identify how turbine efficiency would be assessed.

Analysis

For an adiabatic one-inlet, one-outlet turbine,

w_out = h_1-h_2 + (V_1^2-V_2^2)/2.

Using SI units, the kinetic-energy contribution is

(V_1^2-V_2^2)/2 = (40^2-120^2)/2 = -6400 J/kg = -6.4 kJ/kg.

Therefore,

w_out = h_1-h_2-6.4 kJ/kg.

The increase in exit velocity consumes part of the available enthalpy drop. The shaft power is

W_dot_out = m_dot w_out = 5w_out.

If an isentropic exit state at 0.1 MPa is found from s_2s=s_1, the turbine efficiency is

eta_t = (h_1-h_2 plus the appropriate kinetic-energy correction)/(h_1-h_2s plus the corresponding ideal kinetic-energy correction),

with the exact form selected consistently with the chosen turbine model. In many engineering calculations, kinetic-energy terms are neglected when they are small compared with the enthalpy change; here they should not be omitted without comparison.

Outcome

The calculation yields a direct relationship between tabulated property data and power production. It also demonstrates that the steady-flow energy equation is not merely an enthalpy-difference formula: kinetic-energy changes may materially affect performance when exit velocities are high.

Learning Points

The control volume must include the complete turbine stage, the heat-transfer assumption must be stated, and the sign of the velocity term must be preserved. Efficiency requires comparison with a clearly defined ideal reference process, not merely division by an arbitrarily selected enthalpy difference.

Common Errors, Limitations, and Professional Judgment

A frequent error is applying the closed-system equation Q-W=Delta U to a turbine or compressor. Because mass crosses the boundary, enthalpy and flow energy must be included. Another error is assuming that every steady-flow device is adiabatic. Turbines may lose heat, compressors may require cooling, and valves may exchange heat during long residence times.

Reference levels also require care. Absolute enthalpy values depend on the adopted reference, but enthalpy differences at consistent reference states do not. Property values from different tables or software packages should not be mixed without checking their reference conventions.

Energy balances may be satisfied even when a proposed process is physically impossible. The first law does not impose entropy generation or determine whether heat can flow in a specified direction. Second-law analysis is therefore required for complete assessment. Similarly, a component model may neglect pressure losses, non-uniform flow, transient storage, or heat leakage. The resulting answer is an engineering approximation, not an exact description of every local phenomenon.

Professional analysis also requires transparent reporting. A report should identify the control volume, state assumptions, list property sources, show units, provide uncertainty or sensitivity where relevant, and distinguish measured data from modeled values. In safety-critical systems, independent verification and applicable codes must supplement the energy balance.

Digital Tools and Emerging Practice

Thermodynamic property libraries, process simulators, spreadsheets, and computational-fluid-dynamics tools are widely used to evaluate control-volume balances. Their value depends on correct state specification and physical interpretation. A simulator can calculate an enthalpy, but it cannot decide whether the selected outlet pressure, phase condition, or heat-loss assumption represents the actual equipment.

Modern plant-monitoring systems use measured temperatures, pressures, flow rates, and electrical power to construct online energy balances. Data reconciliation can identify sensor bias and detect abnormal performance. Digital twins may combine first-law models with equipment histories to estimate fouling, leakage, or degradation. These tools should preserve conservation laws, document property models, and expose uncertainty rather than conceal it behind numerical precision.

Best Practices for Engineering Application

The following practices produce robust first-law analyses:

• Define the boundary before selecting equations.
• Write mass and energy balances in general form first.
• Use enthalpy for mass-flow energy transport.
• Separate total quantities from specific quantities and rates.
• State every simplifying assumption and estimate its significance.
• Use consistent units and compatible property data.
• Check signs, limiting cases, and physical plausibility.
• Compare actual performance with a clearly defined ideal reference.
• Report neglected terms when they could influence decisions.
• Use software to support, not replace, thermodynamic reasoning.

Chapter Summary

The first law provides a systematic method for accounting for energy in closed systems and control volumes. Closed systems contain fixed mass and are commonly analyzed using changes in internal, kinetic, and potential energy. Open systems require mass conservation and recognition that flowing matter transports enthalpy, kinetic energy, and potential energy.

The steady-flow energy equation reduces to useful device-specific forms: turbines produce work from enthalpy decrease, compressors and pumps consume work to raise pressure, nozzles convert enthalpy into velocity, throttles operate approximately at constant enthalpy, mixing chambers conserve mass and enthalpy under adiabatic assumptions, and heat exchangers transfer enthalpy between streams. Storage devices require transient mass and energy balances.

Reliable analysis depends on clearly defined boundaries, explicit assumptions, consistent property data, dimensional checks, and physical interpretation. First-law analysis establishes energy feasibility and component performance, but complete thermodynamic assessment may require second-law, mechanical, safety, economic, and environmental considerations.

Key Terms

System: A specified quantity of matter or region selected for thermodynamic analysis.

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

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

Control surface: The boundary enclosing a control volume.

Internal energy: Microscopic energy stored within a substance.

Kinetic energy: Energy associated with macroscopic motion.

Potential energy: Energy associated with elevation in a gravitational field.

Heat: Energy transfer caused solely by a temperature difference.

Work: Energy transfer associated with an organized force–displacement interaction.

Enthalpy: The property h=u+pv, convenient for describing flowing matter.

Flow work: Work required to push fluid across a control surface.

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

Transient process: A process involving time-dependent changes in stored mass or energy.

Throttling: A pressure-reducing process that is approximately isenthalpic.

Isentropic process: An idealized process with constant entropy.

Turbine efficiency: The ratio of actual turbine work to ideal isentropic turbine work under a specified definition.

Compressor efficiency: The ratio of ideal isentropic compression work to actual compression work.

Heat exchanger: A device that transfers heat between streams without necessarily mixing them.

Mass flow rate: The rate at which mass crosses a control surface.

Energy balance: An equation equating net energy transfer to change in stored energy.

Review Questions

Short Answer Questions

1. Why is enthalpy used in the energy balance for a control volume?

2. State the steady-flow energy equation for one inlet and one outlet.

3. What is the principal difference between a closed system and a control volume?

4. Why is a throttling process normally described as isenthalpic?

5. Under what conditions can kinetic and potential-energy terms be neglected?

Long Answer Questions

1. Derive the general energy balance for a control volume and explain the physical meaning of each term.

2. Compare the first-law analyses of turbines, compressors, pumps, and nozzles.

3. Explain the mass and energy balances for a transient filling process in a rigid tank.

4. Develop the energy balance for an adiabatic heat exchanger with two streams and discuss the assumptions involved.

5. Explain how turbine and compressor isentropic efficiencies are defined and interpreted.

Analytical / Application Questions

1. A steady-flow nozzle has negligible inlet velocity, no heat transfer, and no shaft work. Develop an expression for its exit velocity in terms of inlet and exit enthalpies. State when the result may be unreliable.

2. Two fluid streams enter an insulated mixing chamber and leave as one stream. Formulate the mass and energy balances and identify the additional information needed to determine the outlet state.

3. A compressor receives an ideal gas and operates adiabatically. Explain how actual outlet enthalpy, isentropic outlet enthalpy, and compressor efficiency are related, and describe the physical meaning of a lower-than-ideal efficiency.

22. Multiple Choice Questions with Answers

Question 1

For a closed, stationary system in which heat and boundary work are the only energy interactions, the first law is

A. Q-W=Delta U

B. Q+W=Delta H

C. Q-W=Delta H+Delta KE

D. W-Q=Delta U

Correct Answer: A. Q-W=Delta U

Question 2

Why does enthalpy appear in the steady-flow energy equation?

A. It includes entropy generation

B. It combines internal energy with flow work

C. It excludes pressure effects

D. It represents only kinetic energy

Correct Answer: B. It combines internal energy with flow work

Question 3

An adiabatic turbine has negligible velocity and elevation changes. Its specific work output is

A. h2-h1

B. h1+h2

C. h1-h2

D. u1-u2 only

Correct Answer: C. h1-h2

Question 4

For an idealized adiabatic throttling valve with negligible kinetic and potential-energy changes

A. T1=T2 for every real fluid

B. s1=s2

C. h1=h2

D. p1=p2

Correct Answer: C. h1=h2

Question 5

In an adiabatic heat exchanger with negligible kinetic and potential-energy changes, which quantity is conserved for the combined control volume?

A. Total enthalpy flow

B. Temperature of every stream

C. Pressure of every stream

D. Entropy flow without generation

Correct Answer: A. Total enthalpy flow

Question 6

A nozzle operates adiabatically with no shaft work. If the exit velocity is greater than the inlet velocity, then, neglecting elevation change

A. h2 is greater than h1

B. h1-h2 supplies the kinetic-energy increase

C. The enthalpy remains constant

D. The pressure must remain constant

Correct Answer: B. h1-h2 supplies the kinetic-energy increase

Question 7

For an adiabatic compressor with negligible kinetic and potential-energy changes, the specific work input is

A. h1-h2

B. h2-h1

C. u1-u2

D. p1-p2

Correct Answer: B. h2-h1

Question 8

A transient rigid tank is filled through one inlet and has no outlet. Neglecting heat transfer and kinetic and potential-energy changes, the energy entering with the mass is represented by

A. m_in u_in

B. m_in p_in

C. m_in h_in

D. m_in V_in^2

Correct Answer: C. m_in h_in

Question 9

Which statement best describes a first-law energy balance?

A. It alone determines whether a process is reversible

B. It accounts for energy but does not impose all second-law restrictions

C. It requires entropy to remain constant

D. It applies only to steady systems

Correct Answer: B. It accounts for energy but does not impose all second-law restrictions

Question 10

In a two-stream adiabatic mixing chamber with one outlet and negligible kinetic and potential-energy changes, the outlet enthalpy is determined primarily by

A. The arithmetic average of inlet enthalpies

B. The mass-weighted inlet enthalpy balance

C. The higher inlet pressure alone

D. The lower inlet temperature alone

Correct Answer: B. The mass-weighted inlet enthalpy balance