Balance equation

In physics, a balance equation represents the change in a quantity-like variable in a limited volume element as an equation. The change takes place in a physical system enclosed by the system boundary or balance area boundary , the balance area.

The balance equation is an extension of the continuity equation , since no conservation law is required for the quantity-like quantity in the balance equation . In addition to the quantities of a charge and the current known from the continuity equation , source or sink terms can thus occur. These describe the creation or destruction of a quantum of the quantity-like size in the volume element.

Balance equations are used in particular in thermodynamics .

Basics

A balance equation generally consists of storage term , transport quantities and source term or sink term . The storage term contains the size to be balanced. This can be a maintenance size or a non-maintenance size. The transport sizes include transports of the size to be balanced across the system boundary. In the case of a mass balance , transport quantities are, for example, mass flows entering or exiting the system. In the last term, the amount of the balance sheet value is considered which is formed in the system (source term) or destroyed (lower term).

In thermodynamics, total mass and total energy cannot be produced or destroyed in a system, so no source or sink terms can be found in the mass balance and energy balance. The entropy balance, on the other hand, has a source term, since entropy can arise in a closed system.

General equations

The general form for a thermodynamic balance equation is

${\displaystyle \mathrm {d} X=\sum _{i}\mathrm {d} X_{i}^{\mathrm {SG} }+\mathrm {d} X^{\mathrm {Q} }}$.

It is

• ${\displaystyle X}$the balance quantity (e.g. mass , energy or entropy )
• ${\displaystyle X^{\mathrm {SG} }}$ a transport size
• ${\displaystyle \textstyle \sum _{i}\mathrm {d} X_{i}^{\mathrm {SG} }}$the term that outputs the changes in all transport sizes ${\displaystyle i}$
• ${\displaystyle X^{Q}}$ the source or sink term

Balance equation for a continuous process

For a continuously running process (example: thermodynamic cycle ) applies

${\displaystyle {\frac {\mathrm {d} X}{\mathrm {d} t}}=\sum _{i}{\dot {X}}_{i}^{SG}(t)+{\dot {X}}^{Q}(t)}$.

Balance equation for a stationary process

A stationary process (example: power plant turbine in continuous operation) is a continuous process in which the state variables are independent of time. The term becomes zero, so it applies ${\displaystyle \textstyle {\frac {\mathrm {d} X}{\mathrm {d} t}}}$

${\displaystyle 0=\sum _{i}{\dot {X}}_{i}^{\mathrm {SG} }+{\dot {X}}^{\mathrm {Q} }}$.

Examples

For a continuous process, the mass balance is

${\displaystyle {\frac {\mathrm {d} m(t)}{\mathrm {d} t}}=\sum _{i}{\dot {m}}_{i}(t)}$

and the entropy balance

${\displaystyle {\frac {\mathrm {d} S(t)}{\mathrm {d} t}}={\dot {S}}^{\mathrm {SG} }(t)+{\dot {S}}_{\mathrm {irr} }(t)}$.

It is the source term of the balance equation. It describes the increase in entropy inside the system due to irreversible processes such as dissipation . ${\displaystyle {\dot {S}}_{\mathrm {irr} }}$

See also

• Rate equation

Individual evidence

1. ↑ ^{a b c} Peter Stephan, Karlheinz Schaber, Karl Stephan, Franz Mayinger: Thermodynamik. Basics and technical applications. Volume 1: One-component systems. 19th edition, Springer Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-30097-4 , pp. 81–82.
2. ^ Peter Stephan, Karlheinz Schaber, Karl Stephan, Franz Mayinger: Thermodynamik. Basics and technical applications. Volume 1: One-component systems. 19th edition, Springer Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-30097-4 , pp. 89-90.
3. ^ ^{A b} Peter Stephan, Karlheinz Schaber, Karl Stephan, Franz Mayinger: Thermodynamik. Basics and technical applications. Volume 1: One-component systems. 19th edition, Springer Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-30097-4 , p. 189.