**An application of the first law of thermodynamics to a process in which any work terms are negligible.**

#### Closed systems

For a closed system, one that always consists of the same material, the first law is *Q* + *W* = Δ*E*, where *Q* is the heat supplied to the system, *W* is the work done on the system, and Δ*E* is the increase in energy of the material forming the system. It is convenient to treat Δ*E* as the sum of changes in mechanical energy, such as kinetic energy and potential energy in a gravitational field, and of internal energy Δ*U* that depends on changes in the thermodynamic state of the material. (Scientific disciplines do not all use the same sign convention for work, so care is needed when consulting textbooks.)

The net heat input *Q* may have positive and negative components. For example, a satellite in space has no work interaction with its surroundings. It receives heat by radiation from the Sun on some surfaces (positive *Q*) and loses heat by radiation to space from others (negative *Q*). If the net heat input is positive, the internal energy of the satellite increases and its temperature rises; if the heat inputs and outputs are in balance, the internal energy of the satellite does not change and its temperature remains constant. Because the rates at which any changes occur are usually of interest, heat balances are often written in terms of heat flow rates (heat per unit time), sometimes denoted by a dot over the symbol, , so that for a process with negligible work, kinetic energy and potential energy terms , the rate of change of internal energy with time.

The thermodynamic distinctions in terminology between heat, work, and energy are not always applied rigorously. For example, in a heat balance on an electric light bulb, the electrical power is strictly a rate of doing work on the system but might be treated as an electrical heat input, to be balanced against the rate of heat loss by radiation and convective cooling of the bulb by the surrounding air.

#### Open systems

Often it is more convenient to apply the first law or a heat balance to an open system, a fixed region or control volume across the boundaries of which materials may travel and inside which they may accumulate, such as a building, an aircraft engine, or a section of a chemical process plant. Then the first law is expressed by the equation below,

where *W*_{S} is the rate of doing shaft work on the system; is the mass flow rate of any stream entering or leaving the control volume; *h* is the enthalpy per unit mass; *c* is the velocity; *gz* is the gravitational potential for each stream at the point of crossing the boundary of the control volume; and *E* is the energy of all material inside the control volume. When conditions inside the control volume do not change with time, although they need not be spatially uniform, *dE*/*dt* = 0, and the balance equation is known as the steady-flow energy equation. There is also a mass balance equation: the total incoming and outgoing mass fluxes must be equal if material cannot accumulate inside the control volume.

Enthalpy is a thermodynamic property defined by *h = u + pv*, where *u* is the specific internal energy (enthalpy per unit mass), *p* the pressure, and *v* the specific volume. It is used, along with shaft work *W*_{S}_{s}, because the derivation of the first-law equation for a control volume from the more fundamental equation for a closed system involves work terms *pv* that are not available for use outside the control volume. Changes in enthalpy occur because of changes in temperature, pressure, physical state (for example, from liquid to vapor), and changes in chemical state.

For example, for a control volume around a domestic heating boiler, the entering material streams are the fuel and the air needed for combustion and the incoming flow of cold water; the outgoing streams are the combustion products and the hot water or steam. The only heat flow would be a small term for heat losses from the hot casing of the boiler to the surroundings. There might be a small electrical work input to drive a circulating pump for the water. The main terms in this heat balance would be the changes in enthalpy. The enthalpy change from fuel to combustion products is negative and is related to, but not quite equal to, the calorific value of the fuel. * See also: ***Enthalpy**; **Thermodynamic principles**; **Thermodynamic processes**