# Richmann's rule of mixing

The Richmannsche mixing rule is a rule to determine the mixing temperature , which on bringing together two (or more) bodies of different temperature setting. It is named after its discoverer Georg Wilhelm Richmann .

Under the condition that there is no change in the aggregate state and the system of the bodies is closed (in particular only heat exchange between the bodies is possible), the following applies:

{\ displaystyle {\ begin {aligned} Q _ {\ text {submitted}} & = Q _ {\ text {added}} \\ m_ {1} \ cdot \ left (h_ {1} (T_ {1}) - h_ {1} (T_ {m}) \ right) & = m_ {2} \ cdot \ left (h_ {2} (T_ {m}) - h_ {2} (T_ {2}) \ right) \\\ end {aligned}}}

If the specific heat capacities can be assumed to be constant, this can be converted to

{\ displaystyle {\ begin {aligned} m_ {1} \ cdot c_ {1} \ cdot (T_ {1} -T_ {m}) & = m_ {2} \ cdot c_ {2} \ cdot (T_ {m } -T_ {2}) \ end {aligned}}}

The resolved formula according to the mixture temperature is then:

${\ displaystyle T_ {m} = {\ frac {m_ {1} \ cdot c_ {1} \ cdot T_ {1} + m_ {2} \ cdot c_ {2} \ cdot T_ {2}} {m_ {1 } \ cdot c_ {1} + m_ {2} \ cdot c_ {2}}}}$

If the heat capacities are not constant, the above formula with an average heat capacity can be used for component i:

${\ displaystyle {\ bar {c}} _ {i} = {\ frac {\ int _ {T_ {m}} ^ {T_ {i}} c_ {i} (T) dT} {T_ {i} - T_ {m}}}}$

The application of this formula may require an iterative procedure to determine the mixture temperature, since the mean heat capacity is also temperature-dependent.

In which

• m 1 , m 2 stands for the mass of bodies 1 and 2,
• c 1 (T), c 2 (T) stands for the temperature-dependent specific heat capacity of bodies 1 and 2, if applicable ,
• T 1 stands for the temperature of body 1, which gives off heat, i.e. the warmer one,
• T 2 stands for the temperature of body 2, which absorbs heat, i.e. the colder one,
• T m stands for the common temperature of both bodies after mixing,
• h 1 (T) and h 2 (T) stand for the specific enthalpy of bodies 1 and 2

After recognizing the conservation of energy , the mixing rule could be derived from the conservation of thermal energy .

If the two bodies are made of the same material (e.g. a mixture of cold and warm water), i.e. c 1 = c 2 , then the use of the constants c 1 and c 2 in the formula can be dispensed with:

${\ displaystyle T_ {m} = {\ frac {m_ {1} \ cdot T_ {1} + m_ {2} \ cdot T_ {2}} {m_ {1} + m_ {2}}}}$

This formula is a weighted arithmetic mean .