1. Design of Risers and Feeding of Castings
A simplified diagram by putting in references to the equations (1, 2 & 4)
there is no Equation 3, diagram not changed
EQ(1) - Freeze Point Ratio (FPR)
FPR=X
X = (Casting Surface/Casting Volume) / (Riser Surface/Riser Volume)
EQ(2) - Volume Ratio (VR) (Y Axis)
VR=Y=Riser Vol/Casting Vol*
Note: The riser volume is the actual poured volume
EQ(4) - (Freeze Point Ratio) Steel
X=0.12/y *
*The constants are from experiments and are empirical
References - AFS Text Chapter 16; Chastain's Foundry manual Vol 2, Google

2. Volumes, Surface Areas, Castings and Risers...
There are relationships between all these items and values that will help in designing a complete mold that controls progressive solidification, and influences directional solidification to produce castings with minimal porosity and shrinkage defects.
This is by ensuring that the riser(s) are the last to solidify.

3. 4 points about the Riser/Casting Relationship
1 - Risers are attached to the heaviest sections of the casting
2 - Risers are the last to solidify 
3 - A casting that has more than one heavy section requires at least one riser per heavy section
4 - Occasionally the thermal gradient is modified at the mold-metal interface by the introduction of a "Chill" that can better conduct the heat away from the casting and lower the solidification time for that section.

5. Gating / Runner Design
Now a look at the flow characteristics of the metal as it enters the mold and how it fills the casting.
Of the flow characteristics
fluidity/viscosity plays a role. Also,
velocity,
gravitational acceleration & vortex,
pressure zones,
molten alloy aspiration from the mold and
the momentum or kinetic energy of a fluid.

6.  The demarcation point is Re < 2000 is considered a Laminar Flow Re > 2000 is considered a Turbulent Flow Objective is to maintain Re below 2000.

7. LAMINAR FLOW- REFERENCE

8. TURBULENT FLOW- REFERENCE

9. SEVERELY TURBULENT FLOW

10. Basic Components of a Gating System
The basic components of a gating system are:
Pouring Basin,
Sprue,
Runners and
Gates that feed the casting.
The metal flows through the system in this order.
Some simple diagrams to be familiar with are:

11. Pouring Basin - This is the "Crucible -Mold Interface", A pouring cup and pouring basin are not equivalents, The pouring cup is simply a larger target when pouring out of the crucible, a Pouring Basin has several components that aid in creating a laminar flow of clean metal into the sprue. The basin acts as a point for the liquid metal to enter the gating system in a laminar fashion.
"Crucible-Mold Interface" is where the metal from the crucible first contacts the mold surface. This area is lower than where the Mouth of the Sprue is located, by having a pool of metal from the flow will be less chaotic than pouring from the crucible down into the sprue. "Dross-Dam" - to skim or hold back any dross from the crucible or what accumulated through the act of pouring. As the lower portion fills and the metal is skimmed, the clean(er) metal will rise up to meet the opening of the sprue in a more controlled fashion.

12. Sprue Placement and Parts
The sprue is the extension of the sprue mouth into the mold
The choke or narrowest point in the taper is the point that would sustain a "Head" or pressure of molten metal.
To reduce turbulence and promote Laminar Flow, from the Pouring Basin, the flow begins a near vertical  incline that is acted upon by gravity and with an accelerative gravity force
Fluids in free fall tend to distort from a columnar shape at their start into an intertwined series of flow lines that have a rotational vector or vortex effect (Clockwise in the northern hemi-sphere, and counter clockwise in the southern hemi-sphere)...

18. Common Ratio's noted are;
Pressurized - is a system where the gate and runner cross-sectional areas are either equal or less than the choke cross-sectional area; A1= Choke = 1 unit A2 = 1st Runner c/s Area = 0.75 unit A3 = 2nd Runner c/s Area = 0.66 unit A4 = 1st Gate = 0.33 unit A5 = 2nd Gate = 0.33 unit
Unpressurized - The key distinction is that the Runner must have a c/s area greater than the Choke, and it would appear that the Gate(s) would equal or be larger than the Runner(s).
Common Ratio's noted are;
1 : 2 : 4; : 3 : 3
1 : 4 : 4; : 4 : 6

19. The rotational effect, though not a strong force, is causing the cork-screwing effect of the falling fluid. If allowed to act on the fluid over a great enough duration or free fall the centrifugal force will separate the flow into droplets.
None of the above promotes Laminar flow, plus it aids the formation of dross and gas pick-up in the stream that is going to feed the casting.

20. Some dimensioning ratio's from Chastain's Foundry Manual (no.2)
By creating a sprue with a taper, the fluid is constrained to retain it's shape, reducing excessive surface area development (dross-forming property) and gas pick-up.
The area below the sprue is the "Well". The well reduces the velocity of the fluid flow and acts as a reservoir for the runners and gates as they fill. 
Some dimensioning ratio's from Chastain's Foundry Manual (no.2)
1- Choke or sprue base area is 1/5th the area of the well.
2- The well depth is twice the runner depth.
3- the Runner is positioned above the midpoint of the well's depth

21. The Runner System
The runner system is fed by the well and is the path that the gates are fed from.
This path should be "Balanced" with the model of heating or AC ductwork serving as a good illustration. The Runner path should promote smooth laminar flow by a balanced volumetric flow, and avoiding sharp or abrupt changes in direction.
The "Runner Extension" is a "Dead-End" that is placed after the last gate. The R-Ext acts as a cushion to absorb the forward momentum or kinetic energy of the fluid flow. The R-Ext also acts as a "Dross/Gas Trap" for any materials generated and picked-up along the flow of the runner.
An Ideal Runner is also proportioned such that it maintains a constant volumetric flow through virtually any cross-sectional area. In the illustration, notice that the runner becomes proportionally shallower at the point where an in-gate creates an alternate path for the liquid flow. 

22. The Gating System
The Gates (in this case) accommodate a directional change in the fluid flow and deliver the metal to the Casting cavity.
Again, the design objective is to promote laminar flow, the primary causes of turbulence are sharp corners, or un-proportioned gate/runner sizes.
The 2 (two) dashed blue areas when added together form a relationship to the dashed blue area of the Runner, which forms a relationship to the Choke or base of the Sprue Area.

23. The issue of sharp corners (both inner and outer) create turbulence, low & high pressure zones that promote aspiration of mold gases into the flow, and can draw mold material (sand) into the flow. None of this is good... By providing curved radius changes in direction the above effects are still at play but at a reduced level. Sharp angles impact the solidification process and may inhibit "Directional Solidification" with cross-sectional freezing...
The image to the right is just too good a representation to pass-up..
By proportioning the gating system, a more uniform flow is promoted with near equal volumes of metal entering the mold from all points. In an un-proportioned system the furthest gates would feed the most metal, while the gates closest to the sprue would feed the least.
(this is counter to what one initially thinks).

24. DIRECTIONAL SOLIDIFICATION-

25. Formulae, Ratios and Design Equations
What is covered so far is comprehensive, and intuitive on a conceptual level, but the math below hopefully offers some insight into quick approximations for simple designs, and more in-depth calculations for complex systems.
Computerized Flow Analysis programs are used extensively in large Foundry operations.
From basic concepts, designing on a state of the art system shall be attempted:
Continuity Equation –
This formula allows calculation of cross-sectional areas, relative to flow Velocity and Volumetric flow over unit time. This is with the assumption that the fluid flow is a liquid that does NOT compress (that applies to all molten metals).

26. Here, a flow passes through A1
(1" by 1", 1 sq")
The passage narrows to a cross-sectional area A2
(.75" by .75", sq")
The passage expands to a cross-sectional area A3
(1" by 1", 1 sq").
Q= Rate of Flow
(Constant - uncompressible)
V=Velocity of flow
A=Area (Cross-section)
If A1 and A2 are considered, the Area A2 is almost half of A1, thus the velocity at A2 has to be almost double of A1.

27. GATING RATIO is- Areas of Choke : Runner : Gate(s)
The base of the Sprue and Choke are the same.
The ratios between the cross-sectional Area can be grouped into either Pressurized or Unpressurized.
Pressurized: A system where the gate and runner cross-sectional areas are either equal or less than the choke cross-sectional area.

28. This example would resolve to a pressurized flow of 1 : 0.75 : 0.66
A1= Choke = 1 Sq Inch
A2 = 1st Runner c/s Area = 0.75 Sq Inch
A3 = 2nd Runner c/s Area = 0.66 Sq Inch
A4 = 1st Gate = 0.33 Sq inch
A5 = 2nd Gate = 0.33 Sq Inch
Areas A2 & A3 do not get added as they are positioned in line with each other and flow is successive between the points and not simultaneous.
While Areas A4 & A5 are added together as flow does pass through these points simultaneously.
This example would resolve to a pressurized flow of 1 : 0.75 : 0.66

29. Unpressurized:
The key distinction is that the Runner must have a cross sectional area greater than the Choke, and it would appear that the Gate(s) would equal or be larger than the Runner(s).
Common Ratio's noted in Chastian's Vol 2 are:
1 : 2 : 4
1 : 3 : 3
1 : 4 : 4
1 : 4 : 6

30. An exception is noted in Chastain with a 1 : 8 : 6 ratio to promote dross capture in the runner system of Aero-Space castings.
The Continuity Equation is simplified with the use of ratios as the velocity is inversely proportional between any 2 adjacent ratio values. ie H : L equates to an increase in velocity while a L : H equates to a drop in velocity.
Laminar Flow is harder to control at a high velocity than a relatively lower velocity.
Chastain's Vol 2 has much more mathematical expressions and calculations.