Enthalpy formulation and heat transfer equation

The energy equation comes from the first law of thermodynamics applied to a fluid element: the rate of change of energy equals the heat added plus the work done on the element.

Enthalpy form

With specific enthalpy $h=e+p/\rho$, the transport equation is

$$\rho\frac{Dh}{Dt}=\frac{Dp}{Dt}+\nabla\cdot(k\nabla T)+\Phi+S_h$$

that is

$$\rho\left(\frac{\partial h}{\partial t}+\vec V\cdot\nabla h\right)=\frac{\partial p}{\partial t}+\vec V\cdot\nabla p+\nabla\cdot(k\nabla T)+\Phi+S_h$$

For an ideal gas $dh=c_p\,dT$. For an incompressible liquid $dh=c\,dT+dp/\rho$, and the $dp/\rho$ part cancels the $Dp/Dt$ term. Either way, the equation can be written in terms of temperature.

Simplification

For incompressible flow with constant $\rho,\ c_p,\ k$ and negligible viscous dissipation and pressure work (low speed):

$$\rho c_p\left(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)=k\nabla^2T+\dot q$$

or

$$\frac{\partial T}{\partial t}+\vec V\cdot\nabla T=\alpha\nabla^2T+\frac{\dot q}{\rho c_p},\qquad \alpha=\frac{k}{\rho c_p}$$

This is the convection-diffusion equation for temperature.

Special cases

• No flow ($\vec V=0$): $\partial T/\partial t=\alpha\nabla^2T$, the transient heat conduction equation.
• Steady, no flow, no source: $\nabla^2T=0$, the Laplace equation.
• Steady 1D conduction with source: $k\,d^2T/dx^2+\dot q=0$.

The ratio of convection to conduction is the Peclet number $Pe=\rho c_pVL/k=Re\,Pr$.