I think the focus on the duct size and length is misleading here. Duct size can matter if it is a severe restriction in the system, but a factor more likely to be a problem is that you are using a axial fan and not an centrifugal fan.
tl;dr
- manufacturers specify flow rate for range hood fans at zero static pressure
- at the flow you need, the fan will encountered a non-trivial amount of static pressure
- axial fans do not achieve high flow rates other than at very low static pressures
- accordingly, the manufacturer's specified flow rate for axial range hood fans is not a good indication of the flow rate you will experience once it is installed
- centrifugal fans do achieve decent flow rates at static pressures typical in range hood installations and their flow rate once installed will be close but still lower than the manufacturer's specified flow rate
- select a centrifugal fan for your range hood if flow rate is a concern
Your Installation's Resistance Curve
Each installation of a range hood will have its own resistance to the flow of air. This will depend on the restriction of the entire air flow circuit. The most common sources of restriction in that circuit are the following:
- the make-up air path into the suite
- the grill or screen in the air path above your stove
- the fan assembly itself
- the ductwork between the hood and the outdoors
- the grill and backflow preventer where the duct exhausts outdoors
- wind that opposes or aids the flow of air from the make-up air to the exhaust locations on the outside of the house (this often happens when the two are on opposite sides of a house's exterior)
Resistance curves plot the air pressure the fan will encounter versus the air flow that the fan is pushing. A characteristic resistance curve follows an affinity law and is therefore parabolic in shape.
SC1 and SC2 in Fig1 are examples of resistance curves. The two different curves emerge when something about the resistance of your installation to air flow changes. For example, SC1 might be the resistance curve with a kitchen window open (less pressure required for the same amount of flow) and SC2 might occur with all windows close (it would take more pressure to achieve the same amount of flow).
Fan Curves
For a given RPM, each fan has its own relationship between the flow it produces and the pressure that has to overcome. These are shown in Fig1 as curves N1 and N2. The RPM of N1 is higher than N2 and, accordingly, it produces more flow for the same amount of pressure.
Determining the Flow Rate
The actual flow rate achieved is found at the intersection of your installation's resistance curve and your fan's curve. So, in our example, with the windows closed and the fan at the higher RPM, this is the intersection of N1 and SC2 indicated on the plot as "B". In HVAC engineering terminology, this intersection is called the "operating point".
How Air Flow is Specified for Range Hoods
Air flow of range hoods is specified at zero static pressure. In Fig1, this corresponds to the flow rate where N1 and N2 cross the x-axis. The specification provided by the manufacturer is not the air flow you will achieve once the fan is installed because there is no real-world situation where the static pressure encountered by the fan is zero.
The Type of Fan Substantially Affects the Fan Curve and Real-World Flow
You are unlikely to be able to obtain either a fan curve for a given range hood nor a resistance curve for your installation. This is not a practical problem in selecting a fan. That is because range hood fans are one of two general fan types each with a dramatically different capability.
Centrifugal Fans
Curves N1 and N2 are typical of a centrifugal fan in that they are able to create meaningful airflow despite static pressure encountered in the installation.
Axial Fans
Fig2 shows a third fan curve A1 which is typical of an axial fan in that it stops producing any meaningful flow at very low static pressures.
Qualitative Comparison
Note that if you were to look at the manufacturer's specifications for the centrifugal fan that has curves N1 and N2 it would be in the same ballpark as the axial fan with curve A1.
But look what happens when these two fan types encounter the resistance of your system with the windows closed (SC1). The operating point of the centrifugal fan at the high RPM is shown at "B" while the operating point of the axial fan is shown at "D".
Note that the air flow of the axial fan when installed is only about 1/3 of the centrifugal fan despite the manufacturer's specifications for airflow being comparable.1
Note also that the airflow at the operating point is only about 1/4 of the manufacturer's specification for the axial fan, while it is more the 3/4 for the centrifugal fan.1
Epilogue: The Oft-Forgotten Makeup Air Path
Remember, each cubic foot of air that you exhaust has to be made up somewhere else in the house. This might be an open window, your furnace's makeup path, the exhaust gases of a fireplace - yikes!, a backflowing bathroom exhaust fan, sewer gases sucked out of air admittance valves, etc. When you have an operating centrifugal range hood fan specified at several hundred or a thousand CFM, you should have a good source of make-up air. Otherwise, you might end up with unhealthy air from unexpected places being sucked into your home.
1 These ratios are for this particular hypothetical graph. They are meant to be instructive about the magnitude of the difference between actual and manufacturer's specified flow rates. By no means are these specific ratios generally applicable. What is generally applicable, however, is that the flow rate for axial range hood fans drops off significantly faster with static pressure than for centrifugal fans.
Calculating Cubic Feet
The first step in determining what size exhaust fan is needed, is to calculate the volume of the room. To do this, you'll simply multiply the length of the room times the width of the room time the height of the room.
- Length = 10 ft.
- Width = 8 ft.
- Height = 8 ft.
10 ft. * 8 ft. * 8 ft. = 640 ft.3
Calculate Equivalent Duct Length (EDL)
The next step is to measure the length of the duct run, and then apply some additional factors to determine the equivalent duct length of the run. For example, if we had this situation.
- Duct type = Insulated Flex.
- Duct length = 15'.
- Duct diameter = 4".
- Number of elbows = 2.
- Number of wall caps = 1.
We'll use these numbers, and the chart below to determine the EDL.
So the example above would look like this.
15' of 4" insulated flex duct x 1.5 = 22.5'
4" elbow adds 15' x 2 = 52.5'
1 4" roof cap adds 30' = 82.5'
Which means in our example, the EDL is 82.5'.
Determine Required CFM
Exhaust fans are sized using Cubic Feet Per Minute (CFM), so you'll have to use the two values calculated above to determine the size of the required fan. To do this, you'll use the chart below (you'll always round up).
Using our example numbers, we'll place our left finger on 640 cu. ft.. Then place our right finger on 90 ft. (because we have to round up). When we slide our left finger to the right, and our right finger down. we'll see that we need a fan rated for at least 150 cubic feet per minute.
Running The Fan Long Enough
Selecting an appropriately sized fan, is not the only thing to worry about. You also want to make sure that the fan runs for long enough, so that the moisture can be completely exhausted. It's often recommended to run the fan for between 20 - 30 minutes, after a shower. This is often made easier by installing an in wall timer to control the fan.
When selecting a timer, make sure it's rated for motor (inductive) loads and does not say "Incandescent only".
Everybody Wants It Quiet
In my opinion, getting the hot moist air out of the building is the highest concern. Though it would seem, some tend to like it quiet while the air is cleared.
You'll find that exhaust fan loudness is measured in Sone, where the lower the value the quieter the fan. If quiet is important to you, you'll want a fan at or below 1-2 sone. Here is a chart from Panasonic, that might help put sones into perspective.
Best Answer
The small duct will not make your fan "useless". However, the fan will not perform to its published air handling or noise specifications. Further, its lifetime may (or may not) be diminished and its warranty might be voided. Buy from a seller with a liberal return policy, try it, and return if you are not satisfied with the result.
Another consideration: You could replace the duct with the appropriate size for its entire length EXCEPT for the point where it penetrates the masonry. Although not ideal, this will be much better than with the undersized duct because of the duct's length. Finally, performance will be even better if the duct is replaced with one that is oversized. This is probably good practice anyway due to its fairly extreme length and this will partially compensate for the flow restriction at the masonry penetration.