Integral energy equation

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The first law of thermodynamics for a fixed-mass system (closed system) states that

{\left. {\frac{{d\hat E}}{{dt}}} \right|_{system}} = \frac{{\delta Q}}{{dt}} - \frac{{\delta W}}{{dt}}     \qquad \qquad(1)

where Q is positive if heat is transferred into the system, and W is positive if the work is done by the system to the surrounding. The mass of system can store energy internally in a number of different forms. Therefore, the total energy \hat E is due to internal energy, E, kinetic, potential, electromagnetic, surface tension and other forms. For the development of equations in this chapter, the contributions of internal and kinetic energies are considered. Other contributions can also be added as a source term or boundary condition based on the physical model.

For a control volume including only one phase, the energy equation for the control volume can be obtained by setting the general property as \Phi  = E + m{\mathbf{V}}_{rel}^2/2 + m{\mathbf{g}}z and \phi  = e + {\mathbf{V}}_{rel}^2/2 + {\mathbf{g}}z in eq.

{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} }

from transformation formula i.e.,

{\left. {\frac{{d\hat E}}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2} + {\mathbf{g}}z} \right)dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2} + {\mathbf{g}}z} \right)dA} }     \qquad \qquad(2)

where E and e are, respectively, internal energy and specific internal energy, and {\mathbf{V}}_{rel}^2 = {{\mathbf{V}}_{rel}} \cdot {{\mathbf{V}}_{rel}}.

Substituting eq. (2) into eq. (1), an integral form of the energy equation may be written as

\frac{{\delta Q}}{{dt}} - \frac{{\delta W}}{{dt}} = \frac{\partial }{{\partial t}}\int_V {\rho \left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2} + {\mathbf{g}}z} \right)dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2} + {\mathbf{g}}z} \right)dA} }     \qquad \qquad(3)

The heat flow, δQk / dt, is attributed to conduction (and radiation, for the case of participating medium) across the boundary and/or to internal generation, i.e.,

\frac{{\delta Q}}{{dt}} = \int_A { - {\mathbf{q''}} \cdot {\mathbf{n}}dA + \int_V {q'''dV} }

where {\mathbf{q''}} is the heat flux vector at the control volume surface, which can be caused by temperature or concentration gradients, as indicated in {\mathbf{q''}} =  - {\mathbf{k}} \cdot \nabla T from the Fourier's law. The dot product of the heat flux vector {\mathbf{q''}} with the unit normal vector n gives heat conducted out of the control volume. The term {\mathbf{q'''}} is the internal heat generation per unit volume, and it can be caused by chemical reaction, electrical heating, etc.

There are four types of work included in the work rate term. The first is shaft work δWsh / dt, which is done by the control volume to its surroundings that could cause a shaft to rotate or raise a weight through a distance. The second type is the work due to body force, δWb / dt. The third term is associated with pressue change, δWp / dt, and the remaining part of work is necessary to overcome viscous force due to normal and shear stress, δWs / dt.

\frac{{\delta W}}{{dt}} = \frac{{\delta {W_{sh}}}}{{dt}} + \frac{{\delta {W_b}}}{{dt}} + \frac{{\delta {W_p}}}{{dt}} + \frac{{\delta {W_s}}}{{dt}}     \qquad \qquad(4)

The work done by the body force is

\frac{{\delta {W_b}}}{{dt}} =  - \int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} } \right]}  \cdot {{\mathbf{V}}_{rel}}dV     \qquad \qquad(5)

The work associated with pressure can be written as

\frac{{\delta {W_p}}}{{dt}} =  - \int_A {p({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}}) \cdot dA}      \qquad \qquad(6)

The viscous work is evaluated by taking the scalar product of the force acting on the surface of the control volume per unit area, {{\mathbf{n}}_k} \cdot {{\mathbf{\tau }}_k}, and the velocity, {{\mathbf{V}}_{k,rel}}, over the entire surface area of the control volume:

\frac{{\delta {W_s}}}{{dt}} =  - \int_A {({\mathbf{n}} \cdot {\mathbf{\tau }}) \cdot {{\mathbf{V}}_{rel}}dA}      \qquad \qquad(7)

Substituting eqs. (4) and (6) into eq. (3), one obtains

\frac{{\delta Q}}{{dt}} - \frac{{\delta {W_{sh}}}}{{dt}} - \frac{{\delta {W_b}}}{{dt}} - \frac{{\delta {W_s}}}{{dt}}

 = \frac{\partial }{{\partial t}}\int_V {\rho \left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2} + {\mathbf{g}}z} \right)dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2} + \frac{p}{\rho } + {\mathbf{g}}z} \right)dA} }  
  \qquad \qquad(8)

For a single phase system neglecting the shaft work and including pressure work as part of {\mathbf{\tau '}}, the integral form of the energy equation (3) becomes

\frac{\partial }{{\partial t}}\int_V {\rho \left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2}} \right)dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2}} \right)dA} }
 =  - \int_A {{\mathbf{q''}} \cdot {\mathbf{n}}dA + \int_V {q'''dV} }  + \int_A {({\mathbf{n}} \cdot {{{\mathbf{\tau '}}}_{rel}}) \cdot {{\mathbf{V}}_{rel}}dA}  + \int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} } \right] \cdot {{\mathbf{V}}_{rel}}dV}  \qquad \qquad(9)

where the left-hand side represents the rate of total energy change in the control volume and the rate of total energy flow into or out of the control volume. The four terms in the right-hand side represent heat transfer across the boundary of the control volume, internal heat generation, work done by the stress (including normal, shear stress, and pressure at the boundary of control volume), and the work done by the body force.

For control volumes including multiple phases, the energy equation becomes (Faghri and Zhang, 2006)

\begin{array}{l} \sum\limits_{k = 1}^\Pi  {\left[ {\frac{\partial }{{\partial t}}\int_{{V_k}(t)} {{\rho _k}\left( {{e_k} + \frac{{{\mathbf{V}}_{k,rel}^2}}{2}} \right)dV + \int_{{A_k}(t)} {{\rho _k}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k})\left( {{e_k} + \frac{{{\mathbf{V}}_{k,rel}^2}}{2}} \right)dA} } } \right]}  \\ 
  = \sum\limits_{k = 1}^\Pi  {\left[ { - \int_{{A_k}(t)} {{{{\mathbf{q''}}}_k} \cdot {{\mathbf{n}}_k}dA}  + \int_{{V_k}(t)} {{{q'''}_k}dV}  + \int_{{A_k}(t)} {({{\mathbf{n}}_k} \cdot {{{\mathbf{\tau '}}}_{k,rel}}) \cdot {{\mathbf{V}}_{k,rel}}dA} } \right.}  \\ {\rm{         }}\left. { + \int_{{V_k}(t)} {\left[ {\sum\limits_{i = 1}^N {{\rho _{k,i}}{{\mathbf{X}}_{k,i}}} } \right] \cdot {{\mathbf{V}}_{k,rel}}dV} } \right] \\  \end{array}     \qquad \qquad(10)

where the second integral in the bracket on the left-hand side is advection of energy due to mass flow.


Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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