Simplest form of thermodynamic cooling equation? I'm writing a personal project program for simulating a nuclear reactor.  This is not my field of expertise and what I'm struggling the most with is a good equation for cooling.  Newton's law of cooling seems like the simplest but it doesn't seem easy to do with varying coolant flow rates.  I'm also not sure how to simulate that across different temperatures of the coolant.  Any help is greatly appreciated.
 A: I think a rate equation approach which uses some thermodynamic quantities were possible is the best way to model this.
As you pointed out the simplest possible model is Newtonian cooling,
$$\frac{\partial T}{\partial t} = -k \left( T - T_e\right)$$
Where $k=Ah/C$ is the cooling rate, $T$ is the temperature and $T_e$ is the environmental temperature. Here the cooling rate includes some reactor specifications like heat transfer area $A$, heat transfer coefficient $h$ and heat capacity of the fuel $C$. The solution of this equation is an exponential decay to temperature $T_e$. 
To explore how multiple cooling approaches might work simply add terms to this equation. For example, let terms with subscripts 1 correspond to heat lost through the walls of the container (probably naive assumption!) and let subscript 2 correspond to heat extracted using a cooling system in the reactor.
$$\frac{\partial T}{\partial t} = -k_1 \left( T - T_e\right) - k_2 (T - T_c)$$
Where $T_c$ is the temperature of the coolant.
When calculating $k_2$ of the cooling systems you need to use the surface area $A_2$ and heat transfer coefficient $h_2$ (probably quite large). Note that $h_2$ will depend on the flow rate of the coolant and its heat capacity. I'll leave the functional form up to you.
Finally, this a fairly idealised model because it assumes all heat generating nuclear reactions have stopped! You could add a positive heat generation term to account for this.
A: The standard approach is to continue to use Newton's Law of convection ($\dot{Q} = hA (T-T_\infty)$) and to hide all of the actual work behind the scenes in the calculation of a dimensionless parameter called the Nusselt Number. The relevant equation is
 $$h = \overline{Nu} \cdot \frac{k}{L}$$
Where


*

*$h$ is the convection coefficient

*$\overline{Nu}$ is the average Nusselt Number across the whole heat-exchanging surface. If you are assuming that your fluid is inside a pipe while it is heated/cooled, then you can probably assume that the flow is fully developed, and then the Nusselt number is the same everywhere ($\overline{Nu} = Nu$). If the space that your fluid has to move within is very large, then Boundary Layers will develop and cause variation in $Nu$ across the surface; you'll then only be interested in $\overline{Nu}$

*$k$ is the thermal conductivity of the fluid

*$L$ is the length of the fluid's path


$Nu$ is calculated from other dimensionless numbers:


*

*The Reynolds Number $Re_\phi = \frac{\rho v \phi}{\mu}$ (where $\phi$ is any characteristic length, often $x$ or $L$)

*The Prandtl Number $Pr$. This is a property of the fluid which you can look up (or calculate from other properties - see the wikipedia link). In general it will vary with temperature, but for small temperature ranges you can assume that it's approximately constant.


As for how exactly $Nu$ is calculated... empirical correlations. Dozens of them, each specific to a different scenario. You will be interested in forced convection (as opposed to natural convection, where fluid motion is entirely due to buoyancy). You will probably be interested in pipe flow, and that flow will probably be turbulent in many cases of interest. As a start, you might try the Dittus-Boelter equation:
$$
 Nu = 0.0023 Re_D^{4/5} Pr^n
$$
where $n$ is 0.4 if the fluid is heated and 0.3 if the fluid is cooled. Note that in this expression $Re_D = \frac{\rho v D}{\mu}$ and you can assume $\overline{Nu} = Nu$ because the flow is fully developed. 
This means that
$$
 h = 0.0023 \left(\frac{\rho v D}{\mu} \right)^{4/5} Pr^n \frac{k}{L}
$$
Which simplifies to
$$
 h = B \cdot v^{4/5} \cdot Pr^n
$$
where $B$ is some constant.
