When calculating the minimum size of a range hood, there are three things you should consider. The size of the cooking surface, the amount of heat produced by the cooking surface, and the volume of the kitchen.
If the range hood is attached to a wall, you should have 100 cubic feet per minute(cfm) per linear foot. So if you have a 30" wide range, you should have a hood rated at 250 cfm ((30/12)*100 =250). If the hood is over an island, you'll use 150 cfm/linear foot. In this case that same 30" cook top, would require 375 cfm ((30/12)*150 = 375).
Next we'll determine the minimum capacity based on British thermal units(BTU)/hour, by dividing the BTU/hour by 100. For example, if we had a cooktop that produced 40,000 BTUs, we would need 400 cfm. If you are using an electric range (measured in watts), simply multiply watts by 3.41214163 to determine BTU/hr.
The final calculation, will be based on the size of the kitchen. The air in the kitchen should be cycled 15 times per hour, so our formula will be ft³/4. If we have a 10ft x 10ft x 8ft kitchen, (10 X 10 X 8)/4 = 200 cfm.
We'll then choose the largest from these three calculations, and that will be the minimum size hood we need. If you are doing more cooking than the average person, or just want a little more air movement. You can always get a larger hood, this is just the minimum size you should consider.
International Residential Code (IRC), says the minimum intermittent exhaust rate for a kitchen is 100 cfm, while the minimum continuous exhaust rate is 25 cfm.
M1507.4 Local exhaust rates. Local exhaust systems shall be designed to have the capacity to exhaust the minimum air flow rate
determined in accordance with Table M1507.4.
So you'll want to make sure the hood is at least capable of achieving these flow rates.
Are you going to run your kitchen hood exhaust through your HRV? I don't think that is an approved configuration because of the grease in the exhaust. Let's assume you don't for the 2nd part of this answer.
An HRV tries to recover heat from air exhausted through the HRV and use it to warm incoming air. When running your kitchen hood, no air will be exhausted through the HRV so the HRV will be acting like an open window: fresh air will come in through it but it will not be warmed.
During a cold winter, I assume you would find this situation (cold outside air distributed through your house) to be uncomfortable and undesirable.
All the literature I have seen assumes you want to take the cold outside air, hundreds of CFM, and heat it up to 70F so that you can turn around and blow it outside. Equipment capable of doing this is expensive and it seems wasteful of energy.
In my house, I have two windows right next to the hood and I just open the windows when I use the hood. Outside air comes in, mixes with the cooking fumes and goes right out. Standing in front of the range is a little colder than the rest of the house but at the same time, you have the heat from the range so it's not a big deal in my experience.
The bottom line is to consider if you want to heat your makeup air and if not, bring it in as close to the hood as possible, but in a way that helps capture the cooking fumes and doesn't just bypass the range.
Best Answer
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
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:
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.