Fourier-Kirchhoff convective heat transfer equation

The differential equation of convective heat transfer Fourier - Kirchhoff is the equation of energy transfer in a fluid.

Content

Vector view ^[1]

\rho \cdot c\cdot \left({\partial T \over \partial t}+({\vec {\upsilon }}\cdot {\vec {\nabla }})T\right)=div\left[{\lambda \cdot gradT}\right]+q_{v}+\mu \cdot \Phi -p\cdot div({\vec {\upsilon }})

\rho

{\ displaystyle \ rho}

- function expressing density, units: kg / m ^ 3

$c$ ${\ displaystyle c}$ - function of specific mass heat capacity, unit of measure: J / (kg · K )

$T$ ${\ displaystyle T}$ - temperature function, unit: K

$t$ ${\ displaystyle t}$ - function of time, units: s

$\rho \cdot c\cdot {\partial T \over \partial t}$ ${\ displaystyle \ rho \ cdot c \ cdot {\ partial T \ over \ partial t}}$ - non - stationary term (expresses the non-stationary process of heat transfer )

${\vec {\upsilon }}$ ${\ displaystyle {\ vec {\ upsilon}}}$ Is the fluid velocity vector, m / s

$\rho \cdot c\cdot ({\vec {\upsilon }}\cdot {\vec {\nabla }})T$ ${\ displaystyle \ rho \ cdot c \ cdot ({\ vec {\ upsilon}} \ cdot {\ vec {\ nabla}}) T}$ - convective term (expresses heat transfer during medium motion)

$\lambda$ ${\ displaystyle \ lambda}$ - fluid thermal conductivity, W / ( m ^ 2 · K);

$grad(T)$ ${\ displaystyle grad (T)}$ - temperature gradient, K / m ;

$div[{\lambda \cdot grad(T)}]$ ${\ displaystyle div [{\ lambda \ cdot grad (T)}]}$ - conductive term (expresses heat transfer by thermal conductivity )

$q_{v}$ ${\ displaystyle q_ {v}}$ - source term (expresses the influx / loss of energy under the influence of internal sources / sinks of heat)

$\mu \cdot \Phi$ ${\ displaystyle \ mu \ cdot \ Phi}$ - dissipative term (expresses the heating of the medium during the dissipation of kinetic energy during movement)

$\mu$ ${\ displaystyle \ mu}$ - dynamic viscosity coefficient;

$\Phi$ ${\ displaystyle \ Phi}$ - dissipative function , unit of measure - W

$-p\cdot div({\vec {\upsilon }})$ ${\ displaystyle -p \ cdot div ({\ vec {\ upsilon}})}$ - term of thermal compression / expansion (expresses the change in the energy of the fluid when it is compressed or expanded)

Note

In minimizing the errors of the transition from a vector equation to an equation in a specific curvilinear coordinate system , for example, spherical , vector analysis can help. Disclosure of vector analysis operators, such as nabla , divergence and gradient, in various expressions, for example, $\rho \cdot c\cdot ({\vec {\upsilon }}\cdot {\vec {\nabla }})T$ ${\ displaystyle \ rho \ cdot c \ cdot ({\ vec {\ upsilon}} \ cdot {\ vec {\ nabla}}) T}$ , may not always be intuitive, including, it may depend on which functions to the left and right of it - vector or scalar - and which operators to the left and right of it.

History

Simplifications

Limited ability to accurately describe some real processes.

Scopes of Action

Notes

↑ Bukhmirov V.V. Lectures on heat and mass transfer, part 2 (Russian) // Ivanovo State Energy University. - 2008. - December. - S. 2-3 .

[1] Bukhmirov V.V. Lectures on heat and mass transfer, part 2 (Russian) // Ivanovo State Energy University. - 2008. - December. - S. 2-3 .