# Mass balance (thermodynamics)

The mass balance for this exemplary system is
${\ displaystyle \ textstyle {\ frac {dm (\ tau)} {d \ tau}} = {\ dot {m}} _ {1} + {\ dot {m}} _ {2} - {\ dot { m}} _ {3} - {\ dot {m}} _ {4}}$

The mass balance is a balance equation of thermodynamics and considers the mass flows entering and exiting the system .

## General equation of mass balance

In a closed system , considering the mass is trivial: Since no mass can get over the system boundary, the mass in the system always remains constant. In an open system, however, mass can be exchanged with the environment. These mass flows with the environment are recorded by the mass balance. The following applies to the mass balance of a basic substance

${\ displaystyle {\ frac {dm (\ tau)} {d \ tau}} = {\ dot {m}} _ {on} (\ tau) - {\ dot {m}} _ {off} (\ tau )}$

Where:

${\ textstyle m}$ the total mass within the system boundary
${\ displaystyle {\ dot {m}} _ {a}}$ the mass flow entering over the system boundary
${\ displaystyle {\ dot {m}} _ {from}}$ the mass flow exiting the system boundary
${\ displaystyle \ tau}$ the time.

If there are several incoming or outgoing mass flows, the mass balance is generalized and the sum of the mass flows is considered.

${\ displaystyle {\ frac {dm (\ tau)} {d \ tau}} = \ sum {\ dot {m}} _ {i} (\ tau)}$

Due to the sign convention usually used , incoming mass flows are counted as positive, exiting mass flows as negative.

In the context of the theory of relativity , it is possible that mass is converted into energy within a system. The mass balance of thermodynamics does not consider such effects.

## Mass balance for a stationary process

In the Joule cycle , the mass balances of the compressor, heater, turbine and cooler are independent of time

If there is a stationary process , for example a cyclic process , the change in the total mass is zero. The mass flows are no longer time-dependent, it applies

${\ displaystyle {\ dot {m}} _ {on} = {\ dot {m}} _ {off}}$

or for several mass flows

${\ displaystyle \ sum {\ dot {m}} _ {i} = 0}$.

## Mass balance for mixtures

In the case of a mixture , if there is no reaction , the conservation of the masses of the individual components applies . If the total mass of this mixture is to be balanced, a separate mass balance is drawn up for each component.

## Mass balance in reactions

If a chemical reaction takes place, the conservation of the masses of the individual components no longer applies. However, the mass of the starting materials is equal to the mass of the products . The mass balance now does not balance mass flows, but the total mass of the components. For the mass balance, the stoichiometric coefficients are multiplied by the molar masses of the individual substances. So goes for the reaction

${\ displaystyle \ mathrm {\ nu _ {A} \ A + \ nu _ {B} \ B \ longrightarrow \ nu _ {C} \ C + \ nu _ {D} \ D}}$

the mass balance

${\ displaystyle \ mathrm {m_ {A} \ A + m_ {B} \ B = m_ {C} \ C + m_ {D} \ D}}$

This mass balance is obtained through

${\ displaystyle \ mathrm {\ nu _ {A} M_ {A} \ cdot \ A + \ nu _ {B} M_ {B} \ cdot \ B = \ nu _ {C} M_ {C} \ cdot \ C + \ nu _ {D} M_ {D} \ cdot \ D}}$

It is

${\ displaystyle \ nu _ {i}}$ the respective stoichiometric coefficient
${\ displaystyle m_ {i}}$ the mass of the respective substance
${\ displaystyle M_ {i}}$ the molar mass of the respective substance.

## Individual evidence

1. ^ 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 , pp. 89-91.
2. Klaus Lucas: Thermodynamics. The laws of energy and matter conversion. 7th corrected edition, Springer Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-68645-3 , pp. 115-131.