1-D Area-Averaged Navier-Stokes Equations

The Thermal Hydraulics Module (THM) solves for conservation of mass, momentum, and energy with 1-D area averages of the Navier-Stokes equations,

t(Aρf)+x(Aρfu)=0 , \frac{\partial}{\partial t}\left(A\rho_f\right)+\frac{\partial}{\partial x}\left(A\rho_fu\right)=0\ ,(1)

t(Aρfu)+x(Aρfu2+AP)=P~Axf2DhρfuuA \frac{\partial}{\partial t}\left(A\rho_fu\right)+\frac{\partial}{\partial x}\left(A\rho_fu^2+AP\right)=\tilde{P}\frac{\partial A}{\partial x}-\frac{f}{2D_h}\rho_fu|u|A(2)

t(AρfEf)+x[Au(ρfEf+P)]=Hwaw(TwallTbulk)A \frac{\partial}{\partial t}\left(A\rho_f E_f\right)+\frac{\partial}{\partial x}\left\lbrack Au\left(\rho_fE_f+P\right)\right\rbrack=H_wa_w\left(T_\text{wall}-T_\text{bulk}\right)A(3)

where xx is the coordinate along the flow length, AA is the channel cross-sectional area, ρf\rho_f is the fluid density, uu is the xx-component of velocity, P~\tilde{P} is the average pressure on the curve boundary, EfE_f is the fluid total energy, ff is the friction factor, HwH_w is the wall heat transfer coefficient, awa_w is the heat transfer area density, TwallT_\text{wall} is the wall temperature, and TbulkT_\text{bulk} is the area average bulk fluid temperature.