Please review the first dozen pages or so of Greenheck's guide for general principles of kitchen venting. http://www.greenheck.com/media/pdf/otherinfo/KVSApplDesign_catalog.pdf. Although this document addresses commercial kitchens, the principles addressed apply to residential pseudo-Pro cooking installations.
You will find on page 9 that electric and gas char broilers require more specific air flow (feet per minute air velocity equals CFM per square foot of hood entry aperture) than general cooking does. If your hood is only as large as your cooktop, and say 24 inches of front-to-back entry aperture distance, then the opening is 8 square feet, and it would be appropriate to achieve 8 sq. ft x up-to-185 CFM/sq. ft. This value is nearly 1500 CFM by itself, and doesn't account for the fan curve reduction in flow pulling air past the baffles and other pressure losses.
Note that the commercial units in the table of Greenheck's Figure 4 may be larger than your range's broiler, as will be the hood sizes, and depending on the unstated size relative to the broiler under consideration, may allow operation of a residential gas char broiler at less the 185 ft/min value. But it won't be as low as the 85 value in the Medium column.
Silencers come from Fantech, and for a 10-inch duct are 14-inches in diameter and 3 ft long.. They won't fit in a wall between a hood and an exterior wall blower or wall cap. They need some duct length.
It is a bad idea to dump the cooking effluent into an outside sitting area. You will get grease on everything in spite of conventional hood baffles or VaH magic lung blowers throwing the grease particulates at the hood structures hoping for impingement.
VaH blowers are at best rated for their behavior in the hood, which doesn't for them have separate baffles. However, if significant ducting or MUA pressure losses are present, their claimed 1.5 X equivalent performance erodes faster than conventional blowers defined by their fan curves. Thus each option needs to be separately analyzed.
I think for this relatively extravagant residential kitchen appliance, a more commercial approach to venting will be needed, and the vented air needs to be eliminated at a roof line using a down-slope centrifugal blower such as Wolf or Broan supply, or preferably upwards using a modest sized up-blast blower. This may require use of an outside chase to get out of the patio area.
Generally, a remote blower with an in-line silencer will have the lowest noise at the cook's ear.
I would like to see the size of the air exchange module that can handle 1500 CFM of MUA. Normally, the MUA is supplied either passively with limited heating and filtering, or with its own blower and control system. MUA issues can be researched here in many MUA related threads. I would just note that dealing with the MUA aspect is often more complex than the venting aspect.
Q