# Control volume In [[Fluid_dynamics|fluid dynamics]] and [[Thermodynamics|thermodynamics]], a **control volume** is a region of space — fixed, moving or deforming — chosen for analysis and bounded by an imaginary surface, the control surface, across which matter is free to pass. Balances of mass, momentum and [[Energy|energy]] are then written for what crosses that surface rather than for a fixed parcel of matter. The choice is a bookkeeping decision, not a physical one: a [[Nozzle|nozzle]], a [[Turbine|turbine]] stage, a stretch of river, a radiator or an entire [[Fossil_fuel_power_station|power station]] can each be wrapped in a control surface, and [[Conservation_of_mass|conservation of mass]] and [[Conservation_of_energy|conservation of energy]] hold for the region so drawn. The alternative bookkeeping, a **control mass** or closed system, follows the same matter wherever it goes and lets the boundary move with it.[^yan-ch1] In the microsim below the reader picks a device — nozzle, diffuser, turbine, compressor, throttle or [[Heat_exchanger|heat exchanger]] — and sets the inlet temperature, [[Pressure|pressure]] and velocity together with the outlet pressure. One equation answers, the steady-flow energy equation q − w_s = Δh + Δ(V²/2) + g·Δz, where q is heat added per kilogram of flow, w_s the shaft work taken out per kilogram, and h the [[Enthalpy|specific enthalpy]]. The readouts give exit velocity, shaft work, heat exchanged and isentropic efficiency. Every device on the selector is that same equation with different terms struck out, and the point of the sim is to watch which terms survive.[^yan-ch5] On the [[Energy]] flagship this article is the child of Part VII — Energy transfer, section *Open systems* (row E61), the C07 embed shared with [[Thermodynamic_system|thermodynamic system]] and with the [[First_law_of_thermodynamics|first law]]. Its sibling rows carry the closed-system form of the same law; this one is what happens when the boundary leaks on purpose. ## Overview Control-volume analysis is a three-step habit: draw the surface, count the mass that crosses it, then count the energy that rides along with that mass. The three steps are taken in that order because each one constrains the next, and because the second step is what forces the third to be written in terms of enthalpy rather than internal energy. What follows walks the steps in order and then applies them to the six devices the microsim offers. #### The system and the control surface Thermodynamics begins by drawing a boundary. Everything inside is the system, everything outside the surroundings, and the boundary is what the analyst chooses to account across.[^yan-ch1] A closed system — a gas trapped under a piston, a sealed refrigerant charge — exchanges heat and work with its surroundings but no matter, so its mass is a constant of the problem and the first law reads ΔU = Q − W. An open system exchanges matter as well, and the region it occupies is the control volume. Nearly every device that does useful work continuously is open: the working fluid streams through it and does not stay.[^yan-ch5][^mitofsky-survey] The control surface may be drawn anywhere, and drawing it well is most of the skill. Put it where the properties are known or uniform — at a pipe flange, at the inlet and exit planes of a [[Steam_turbine|steam turbine]], around both streams of a heat exchanger so that the heat crossing between them never crosses the surface at all. A surface drawn through the middle of a turbulent recirculation buys nothing, because the fluxes across it cannot be evaluated. The same device analysed with two different surfaces gives two different, equally correct sets of books. #### Mass in, mass out The mass balance for a control volume is the statement that mass cannot accumulate unless more arrives than leaves: dm_cv/dt = Σ ṁ_in − Σ ṁ_out, with the mass flow rate ṁ = ρ·A·V through an opening of area A carrying uniform density ρ and normal velocity V. Under **steady flow** — the assumption behind almost every device analysis, meaning no property inside the control volume changes with time — the left side vanishes and Σ ṁ_in = Σ ṁ_out.[^yan-ch5] For a single-stream device that collapses to ρ₁·A₁·V₁ = ρ₂·A₂·V₂, which is the [[Continuity_equation|continuity equation]] in its most familiar form, and it is why a nozzle that squeezes a subsonic gas speeds it up while a [[Compressible_flow|compressible]] supersonic stream does the opposite. Steadiness is an assumption, not a law, and the sim states it plainly. A turbine at constant load is steady; the same turbine during a start-up, or a tank being filled, is not, and then dm_cv/dt is the whole story. #### Why enthalpy appears The energy balance is where the control volume earns its own equation rather than borrowing the closed-system one. Pushing a kilogram of fluid across the inlet plane against the pressure there costs work p₁·v₁, and the fluid leaving does work p₂·v₂ on whatever is downstream. This *flow work* is not shaft work and not heat; it is the price of admission, and it is unavoidable in every open system. Adding it to the [[Internal_energy|internal energy]] u of the stream produces the combination u + p·v, which is precisely the enthalpy h. Enthalpy is therefore not a convenience in control-volume analysis but the natural energy carried by a flowing stream, and this is the reason it appears in every device equation that follows.[^yan-ch2][^yan-ch5] With flow work folded into h, the general energy balance for a control volume is dE_cv/dt = Q̇ − Ẇ_s + Σ ṁ_in·(h + V²/2 + g·z)_in − Σ ṁ_out·(h + V²/2 + g·z)_out, and for one inlet, one outlet and steady flow it becomes the steady-flow energy equation of the lead, q − w_s = Δh + Δ(V²/2) + g·Δz, all terms per kilogram of throughput.[^yan-ch5] For an [[Ideal_gas_law|ideal gas]] with a constant [[Specific_heat_capacity|specific heat]] the enthalpy difference is simply Δh = c_p·ΔT, which is what the sim uses for its air presets, with c_p = 1.005 kJ/(kg·K) and the gas constant R = 0.287 kJ/(kg·K) for air near room temperature.[^yan-appg] Holding c_p constant across a 200 K span is an idealization that costs a few percent; the steam preset avoids it by reading enthalpy from the water tables instead, the same tables that give 3 MPa, 350 °C steam a specific volume of 0.09056 m³/kg where the ideal-gas law would predict 0.09586 m³/kg, an error of 5.9 %.[^yan-ch3][^yan-appa] #### Six devices, one equation Each device on the selector is the steady-flow energy equation with the terms that do not matter set to zero, and the honest work of the analysis is justifying each deletion. | Device | Terms kept | What it does | |---|---|---| | Nozzle | Δh + Δ(V²/2) = 0 | trades enthalpy for speed | | Diffuser | Δh + Δ(V²/2) = 0 | trades speed for pressure | | Turbine | w_s = −Δh | takes shaft work out | | Compressor | w_s = −Δh | puts shaft work in | | Throttle | Δh = 0 | drops pressure, gains nothing | | Heat exchanger | q = Δh per stream | moves heat between streams | *The sim's device presets. Potential-energy change g·Δz is negligible in all six; heat loss is neglected in the first five.*[^yan-ch5] A **nozzle** has no shaft and, being short, little time to lose heat, so q and w_s both drop out and the kinetic term carries everything: V₂ = √(V₁² + 2·(h₁ − h₂)). Air entering at 500 K and 30 m/s and leaving at 400 K therefore leaves at √(30² + 2 × 1005 × 100) = 449 m/s (derived) — a hundred kelvin of temperature bought four hundred metres per second. A **diffuser** is the same algebra run backwards, and it is why the inlet of a [[Jet_engine|jet engine]] is a diffuser before it is anything else. A **turbine** has a shaft, and the kinetic term is small because inlet and exit ducts are sized to keep velocities comparable, so w_s = h₁ − h₂. Air expanding from 700 K to 500 K delivers 1.005 × 200 = 201 kJ/kg, and at ṁ = 10 kg/s that is 2.01 MW of shaft [[Power_(physics)|power]] (derived), since power is energy per unit time and 1 W = 1 J/s.[^murphy-work] A **compressor** is a turbine with the sign reversed and, importantly, is never adiabatic in the same benign way: the work goes in, the temperature rises, and real machines are cooled. A **throttle** — a valve, a porous plug, a capillary — has neither shaft nor time for heat transfer nor room for kinetic energy, so every term vanishes and h₁ = h₂. For an ideal gas whose enthalpy depends only on temperature that means no temperature change at all; for a real fluid near saturation it means a large one, which is how a refrigeration cycle gets cold without any moving part at the expansion end. A **heat exchanger** has no shaft work, and if the control surface is drawn around both streams the heat crossing between them is internal, so the balance is simply ṁ_c·Δh_c = −ṁ_h·Δh_h.[^yan-ch5][^mitofsky-survey] #### Isentropic efficiency and the second law Mass and energy are conserved in every one of these devices; that is what makes the first law weak on its own. It will happily describe a throttle that raises pressure or a turbine that delivers more work than the flow contains, and only the [[Second_law_of_thermodynamics|second law]] rules those out. The reference against which real machines are scored is the isentropic device — one that is [[Adiabatic_process|adiabatic]] and reversible, with no friction, no mixing and no heat transfer across a finite temperature difference.[^yan-ch6] For an ideal gas the isentropic exit temperature follows from T₂s/T₁ = (p₂/p₁)^((γ−1)/γ), with γ = 1.4 for air. Isentropic efficiency then compares real to ideal, in the direction that makes it a number below one: for a turbine η = (h₁ − h₂)/(h₁ − h₂s), for a compressor η = (h₂s − h₁)/(h₂ − h₁). Air compressed from 300 K and 100 kPa to 800 kPa would reach 300 × 8^0.2857 = 543 K and cost 244.6 kJ/kg if the compression were reversible; at η = 0.80 it costs 306 kJ/kg and arrives at 604 K instead (derived). The extra 61 kJ/kg did not vanish — it went into the stream as heat, which is why the real exit is hotter, and it is [[Entropy|entropy]] generation made visible. The sim shows the ideal and actual states as two points and the gap between them as the lost work, which is the same quantity that the [[Carnot_cycle|Carnot]] bound names for a whole cycle.[^yan-ch6] Assembling devices into a cycle is then arithmetic: a [[Rankine_cycle|Rankine]] plant is a pump, a boiler, a turbine and a condenser, each a control volume, wired so that one's outlet is the next's inlet. Large steam plants push the turbine inlet to 300–600 °C precisely because the second law rewards a higher source temperature.[^yan-ch6-steam] Every irreversibility on the list — friction, unrestrained expansion, mixing, heat transfer across a finite ΔT, electrical resistance — is a term the ideal analysis omits and the efficiency then puts back.[^yan-ch6] ## Substantive derivative A control volume is a choice about where to stand. The substantive derivative is the rule that converts between the two possible choices — standing still while the fluid goes past, or going past with the fluid — and it is the reason an integral balance over a fixed region ends up carrying flux terms at all. #### Following the particle, watching the point The control volume is one of two ways to describe a moving fluid, and the substantive derivative — also called the material or convective derivative — is the bridge between them. In the *Lagrangian* description an observer rides with a fluid particle and reports what happens to it. In the *Eulerian* description, the one a control volume uses, observers sit at fixed points and report what passes. The rate of change a rider measures, written D/Dt, and the rate of change a fixed observer measures, ∂/∂t, differ by the rate at which the flow carries the rider into surroundings with different properties: D(·)/Dt = ∂(·)/∂t + (u·∇)(·), with u the velocity field. The first term is *local* or unsteady change; the second is *convective* change, and it is nonzero even in a flow that never changes at any fixed point. A river makes the distinction concrete. Suppose water flows at 2 m/s through a reach whose temperature rises steadily by 0.5 °C per kilometre downstream, and suppose nothing about the river changes from hour to hour. A thermometer bolted to a bridge reads a constant temperature: ∂T/∂t = 0. A thermometer floating with the water warms at DT/Dt = u·∂T/∂x = 2 × 0.0005 = 0.001 °C/s, or 3.6 °C per hour (derived). Both instruments are correct, and neither is measuring what the other measures. The same split explains why a fluid particle can accelerate through a steady nozzle: the velocity at every fixed station is constant, but the particle moves to stations where it is larger. #### From the derivative to the surface integral Integrate the substantive derivative over a region and the convective term turns into a flux through the boundary of that region — the transport theorem that underlies every control-volume balance. In words: the rate of change of any extensive property carried by the matter currently inside the control volume equals the rate of change of that property stored inside, plus the net rate at which the property is carried out through the control surface. Applied to mass it gives the continuity equation of the previous section; applied to momentum it gives the control-volume momentum balance and, in differential form, the [[Navier–Stokes_equations|Navier–Stokes equations]]; applied to energy it gives the steady-flow energy equation the microsim solves.[^yan-ch5] That is the whole relationship between the two sections of this page. The substantive derivative is the pointwise statement, valid at every location in the flow; the control-volume balance is its integral over a region an engineer can actually put a flange on. A [[Boundary_layer|boundary layer]] analysis needs the first; sizing a [[Wind_turbine|turbine]] or a [[Heat_transfer|heat exchanger]] needs the second, and the sim's six devices are six answers the second form gives once four of its terms are argued away. *See also:* [[Enthalpy]] · [[Thermodynamic_system]] · [[Open_system_(systems_theory)]] · [[Turbine]] · [[Nozzle]] ## See also - [[Enthalpy]] - [[Thermodynamic_system]] - [[Open_system_(systems_theory)]] - [[Turbine]] - [[Nozzle]] - [[First_law_of_thermodynamics]] - [[Continuity_equation]] - [[Heat_exchanger]] - [[Rankine_cycle]] ## References [^yan-ch1]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 1, "Basic Concepts and Definitions", pp. 31–58 (system, boundary, surroundings; closed and open systems; page to pin). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^yan-ch2]: Yan (2022), *Introduction to Engineering Thermodynamics*, Chapter 2, "Thermodynamic Properties of a Pure Substance", pp. 59–104 (enthalpy as u + p·v; the property tables; page to pin). Portal Book 115. [^yan-ch3]: Yan (2022), Chapter 3, "Ideal and Real Gasses", pp. 105–126; specifically p. 109–110 (the ideal-gas model holds at high temperature and low pressure, and using it unchecked is "a common mistake"), p. 116 and p. 122 (the compressibility factor Z = p·v/(R·T) and v = Z·R·T/p), and pp. 112, 114 and 122 (gas constants: methane 0.5182, oxygen 0.2598, steam 0.4615 kJ/(kg·K)). Portal Book 115. [^yan-ch5]: Yan (2022), Chapter 5, "The First Law of Thermodynamics for a Control Volume", pp. 187–238 — the mass and energy balances for a control volume and the worked analyses of nozzles, diffusers, turbines, compressors, throttling valves and heat exchangers (page to pin). The equation displays in this book were lost in the text extraction used for the Portal Book index; the forms printed on this page are the standard textbook forms and should be checked against the PDF page. Portal Book 115. [^yan-ch6]: Yan (2022), Chapter 6, "Entropy and the Second Law of Thermodynamics", pp. 239–348; p. 269 lists the irreversibilities (friction, unrestrained expansion, mixing, heat transfer across a finite ΔT, electrical resistance, inelastic deformation, chemical reaction) and p. 272 gives the Carnot efficiency and the second-law inequality that rules out a device above it. Portal Book 115. [^yan-ch6-steam]: Yan (2022), Chapter 6, p. 273 (large steam plants raise the source temperature by running 300–600 °C steam). Portal Book 115. [^yan-appa]: Yan (2022), Appendix A, "Thermodynamic Properties of Water", pp. 349–368 (the steam tables; the 3 MPa, 350 °C entry gives v = 0.09056 m³/kg, against the ideal-gas value of 0.09586 m³/kg computed from R = 0.4615 kJ/(kg·K), an error of 5.9 %). Portal Book 115; the table value and the comparison are also given at 115 pp. 121–123. [^yan-appg]: Yan (2022), Appendix G, "Properties of Various Substances", pp. 413–419 (tabulated specific heats and gas constants; the values used here for air, c_p = 1.005 kJ/(kg·K), R = 0.287 kJ/(kg·K) and γ = 1.4, are standard tabulated values — page to pin). Portal Book 115. [^mitofsky-survey]: Mitofsky, Andrea (2018). *Direct Energy*. Part I, "Survey of Energy Conversion Devices", pp. 33–254 (devices analysed as flow systems with an input and an output stream; page to pin). Portal Book 055, https://open.umn.edu/opentextbooks/textbooks/direct-energy [^murphy-work]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*, pp. 88–91 (work as W = F·d, 1 J = 1 N·m "when the motion is aligned with the direction of force"; power as energy per time, 1 W = 1 J/s; Table 5.2's list of energy forms). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet ### Notes Two conventions on this page deserve a flag. First, the sign of shaft work: w_s is taken positive when work leaves the control volume, so a turbine has w_s > 0 and a compressor w_s < 0; some texts reverse this and every sign in the device table flips with it. Second, the air presets hold c_p constant, which is an idealization rather than a measured law — over the 400–700 K spans used here the true c_p of air rises by several percent, so the worked values above are good to about that much and are marked (derived) where they were computed for this page rather than printed in a source. ## External links - [Introduction to Engineering Thermodynamics](https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics) (Yan, 2022) — Portal Book 115; Chapter 5 is the control-volume chapter - [Direct Energy](https://open.umn.edu/opentextbooks/textbooks/direct-energy) (Mitofsky, 2018) — Portal Book 055 - The Wikipedia pair's external links list further open lecture notes and course pages ### PDFs Both open textbooks above are distributed as free PDFs from their Open Textbook Library records, which is the stable place to fetch them; the Wikipedia pair's own *PDFs* subsection lists the lecture-note scans it prefers, and those are not reproduced here. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Control_volume.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Control volume* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Control_volume.html" data-title="Control volume"></div> *Built from `MICROSIM_GUIDE/specs/sims/Control_volume.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Control_volume) : [Wikitube](https://en.wikitube.io/wiki/Control_volume) · pinned revision [1355018816](https://en.wikipedia.org/w/index.php?oldid=1355018816) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E61 · sim pending (matter/Control_volume).*