Fill rate of tank for elevated vessel 
I'm trying to model a system that consists of a rectangular vessel (storage tank) that fills a cylindrical tank. I would like the compute the time it takes to fill a tank based on an initial vessel water height $H_\text{vw}$.
The vessel tank, seen on the left:


*

*is open to the atmosphere and filled with water

*is rectangular

*has a height of $H_\text{v}$ (constant)

*has an instantaneous water level of $H_\text{vw}$

*is always being filled by a constant flow $F_\text{f}$ whenever the water level  - $H_\text{vw}$ is less than the tank height $H_\text{v}$. $F_\text{f}$ is in gallons per minute, but I can easily convert it to inches of water per minute.

*The bottom of the vessel tank sits $H_\text{ag}$ above the bottom of the fill tank.


The fill tank, to the right:


*

*is open to the atmosphere

*is cylindrical in shape with a diameter $d$

*has an instantaneous water level of $H_\text{tw}$

*has a maximum water height of $H_\text{t}$

*is connected to the vessel tank via a 2" flexible smooth walled hose.


I have a constraint to fill the tank within a certain amount of time $t_\text{max}$, for a given initial water level in the vessel $H_\text{vw}(0)$,g so I need to choose a height $H_\text{ag}$ that makes this possible.
After I solve for $H_\text{ag}$, I would like to know how long it takes to fill a tank for any given initial water level $H_\text{vw}(0)$.
I would also like to model the flow rate $\text{gpm}(t)$ in gallons per minute during the fill process.
 A: Provided that the transfer pipe has a wide bore and is not very long, the flow in that pipe will not be viscous. Flow will be dominated by inertia. 
Torricelli's Law (which can be derived from Bernoulli's Equation) provides a good estimate of the fluid velocity at the "outlet", which is the water surface in the receiving tank. In your setup the head (=difference in height between water levels) is not constant, so the outlet speed is not constant. The cross-section of the cylinder is not constant either, so the head does not vary in a simple, linear manner. An exact model would be tedious to set up and solve.
Nevertheless you could get a good enough estimate by assuming that the water surface in the cylindrical tank remains at the mid level and calculate the head $h$ from this. Then use Torricelli's Law to get the particle flow speed $v$ based on this value of $h$, but use the cross-section area $A$ of the pipe to calculate volume flow rate $Av$ at the outlet level.  
I don't see any point in trying to be accurate. The strength of engineering is being able to make simple estimates for design then experiment with the variables in your apparatus to get the best performance. When your apparatus is built you can vary the head of water easily, if the storage tank is tall enough. Alternatively you can use a valve to control the flow rate through the transfer pipe.
