Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral forms of Momentum and Energy Equations

  1. Sep 1, 2012 #1
    Hi

    I was reading a book that introduced momentum and energy in integral forms and I had some confusion regarding what the terms meant. All integrals below are closed integrals

    For the momentum equation, the result was:

    F = d(mV)/dt = ∫∫ρ(V[dot]dS)V + ∫∫∫∂(ρV)/∂tdV

    From product rule, d(mV)/dt = dm/dt*V + mdV/dt. I can see how dm/dt*V is reflected in ∫∫ρ(V[dot]dS)V but I do not see how mdV/dt is reflected in ∫∫∫∂(ρV)\∂tdV?

    For the energy equation, the result was:

    ∫∫∫qdotρdV - ∫∫pV[dot]dS + ∫∫∫ρ(f[dot]V)dV = ∫∫∫∂(ρ(e+V^2/2))/∂t + ∫∫ ρ(e+V^2/2)V[dot]dS. p is pressure, e is specific internal energy, f is body force per unit mass
    , V is velocity, and qdot is heat transfer per unit mass

    from another book, another form of this equation was:

    Qdot - Wdot = dEcv/dt + mdotout (hout+V^2/2) - mdotin (hout+V^2/2)

    I'm basically wondering, is the mdot*h (the enthalpy term) reflected in the terms ∫∫ ρ(e+V^2/2)V and ∫∫pV[dot]dS or in other words is mdot*(e + V^2/2)-> ∫∫ ρ(e+V^2/2)V and mdot*p*v -> ∫∫pV[dot]dS?

    Thanks
     
    Last edited: Sep 1, 2012
  2. jcsd
  3. Sep 1, 2012 #2

    boneh3ad

    User Avatar
    Science Advisor
    Gold Member

    In the momentum equation, the terms are as follows:
    [tex]\iiint\rho\vec{v} (\vec{v}\cdot d\vec{s})[/tex]
    represents the mass flux across the surface of the control volume.
    [tex]\iint\dfrac{\partial (\rho\vec{v})}{\partial t}dV[/tex]
    is the time rate of change of momentum in the control volume. Usually there is also a body force term and a surface force term (pressure and viscosity if you are doing viscous flows).

    For your energy equation, the terms are as follows:
    [tex]\iint\dot{q}\rho\;dV[/tex]
    is the rate of heat added to the control volume.
    [tex]-\iint p \vec{v}\cdot d\vec{s}[/tex]
    is the rate of work done on the fluid in the control volume by pressure forces.
    [tex]\iiint\rho(\vec{f}\cdot\vec{v})\;dV[/tex]
    is the rate of work done on the fluid in the control volume by body forces.
    [tex]\iiint\dfrac{\partial}{\partial t}\left[ \rho\left( e + \dfrac{|\vec{v}|^2}{2} \right) \right]dV[/tex]
    is the the time rate of change of energy inside the control volume.
    [tex]\iint\rho\left( e +\dfrac{|\vec{v}|^2}{2}\right)\vec{v}\cdot d\vec{s}[/tex]
    is the net energy flux across the surface of the control volume.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral forms of Momentum and Energy Equations
Loading...