OK there seems to be a continued interest in this question so here is more.
Mattlock, Welcome but I'm afraid that your MIT lecture doesn't cover this topic.
Let us start with a control volume
$\displaystyle {U_1} + {P_1}{V_1} + K{E_1} + P{E_1} + Q = {U_2} + {P_2}{V_2} + K{E_2} + P{E_2} + W$
What the given equation is telling us is that as a parcel of fluid passes through this control volume in a time $\displaystyle \partial T$ and in a steady state condition, it looses some energy by conduction. This outflow of energy from the control volume is replenished by new fluid entering the control volume.
This situation is covered by the open system version of the First Law of Thermodynamics, otherwise known as the steady flow equation.
$\displaystyle {U_1} + {P_1}{V_1} + K{E_1} + P{E_1} + Q = {U_2} + {P_2}{V_2} + K{E_2} + P{E_2} + W$
Where U is the internal energy of the entering fluid, PE is the (gravitational) potnetial energy and KE is the kinetic enrgy of the fluid parcel.
Q is the external heat supplied and W is the external work performed.
Now we simplify this as follows:
We assume no change to potential energy
$\displaystyle P{E_1} = P{E_2}$
Since the given equation shows u, v and w constant there is not change to the kinetic energy
$\displaystyle K{E_1} = K{E_2}$
We assume there is no external heat supplied at the boundary walls
$\displaystyle Q = 0$
And that no external work is performed.
$\displaystyle W = 0$
this leaves
$\displaystyle {U_1} + {P_1}{V_1} = {U_2} + {P_2}{V_2}$
But the definition of enthalpy is
$\displaystyle H = U + PV$
or
$\displaystyle \partial H = {U_2} + {P_2}{V_2} - \left( {{U_1} + {P_1}V} \right)$
The equation given uses i = specific enthapy and operates in 3 dimensions so we have the first term in the equation
$\displaystyle \left[ {u\rho \frac{{\partial i}}{{\partial x}} + v\rho \frac{{\partial i}}{{\partial y}} + w\rho \frac{{\partial i}}{{\partial z}}} \right]$
This is equated to the Heat diffusion equation with no sources or sinks in a steady state, which forms the second term in the given equation, after the minus sign.
This equation links the heat flow to the extenal temperature distribution by way of the specific conductivity as already noted.
Because there is a steady state the mass inflow of enthalpy exactly balances the conduction outflow of heat so one the first is considered positive and the second negative and their difference is zero.
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Last edited by studiot; May 12th 2017 at 12:33 PM.
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