Hi there Joshua, and welcome to the forum!
This is probably a bit more complicated than you'd like to hear, but the good news is that there is a solution.
As FriFlo already pointed out, your room does need ventilation, absolutely (because you need to breathe!), so you cannot just block it and do nothing. Not an option. Therefore, you have two other initial choices: Either keep this vent and install a silencer on it, or block this one off then make another vent some other place.
As FriFlo also pointed out, one single vent by itself is useless for a studio: You need two vents: one that brings fresh air into the room, and the other to remove the stale air. Air will only flow through the room if there is a pressure differential to drive it. One single path between the room and the outside cannot possibly create a pressure differential (unless you open a door or window, or the if the room is not sealed air-tight, in which case it has no business being a studio anyway!) So You must have two vents, and also a fan to create the pressure differential. The fan can go anywhere; either the inlet duct or the exhaust duct, but you do need one. Without it, no air will move.
OK, so lets assume that you decide to keep this vent, since it is already there, and will use it as either the inlet or the exhaust. Great. That can be done, provided that you put a silencer on it and also install the other vent at some other location in the room, also with a silencer on it.
So now comes the math you were asking about. The first thing you need to figure out, is how much air you need to have moving through that vent. That is generally calculated using the strange units of "room changes per hour". In other words, how many times per hour you should replace the entire volume of air in the room. There are tables and recommendations for that, for various different types of room, from libraries to hospital operating theaters, bedrooms to factories, cinemas to bathrooms, etc. But a rough rule of thumb for studios, is about 6 to 8. So assume that you need to replace all the air in your room, between 6 and 8 times every hour. You said the dimensions of your room are 17.5' x 10.5' x 7.5', so the volume is roughly 1,400 cubic feet. Times 6 = 8,400, or times 8 = 11,200, so you need to move about 8,000 to 12,000 cubic feet of air through your room every hour. Fans are normally rated in cubic feet per MINUTE, not per hour, so divide by 60: you need a fan that can move about 130 to 190 cubic feet per minute (CFM). Call it 150 CFM, for the sake of argument. (I'm rounding up and down all over the place here, to make the math simpler).
So far so good: but the air does not flow freely through your room system: there is
resistance to the flow of air (which you'll need to calculate at some point) due to the "back pressure" that the fan has to work against. In HVAC jargon, this is called "static pressure", and is expressed in units of "inches of water column". It works sort of like blood pressure. Think of static pressure as being the total resistance that your entire room + ducts + silencers presents to the fan, just like blood pressure is the total resistance the your veins + arteries present to your heart.
Fans are designed to work in a certain static pressure range, so you do have to calculate what it is in your system. The higher the static pressure, the less air the fan can move. So if you bought a fan rated to produce 150 CFM flow volume into a static pressure of 0.5 inches of water column, but the static pressure in your system is actually 0.75, not 0.5, then the fan will not be able to actually move 150 CFM, since the resistance is too high. It will move less. The rating table on the fan data sheet will tell you how much air it can move for each static pressure, which is why you need to know what that static pressure is in your system, so you can buy the right fan. You figure out static pressure by knowing your duct diameter and length, including the silencer boxes. There are tables that show you that for xxx feet of yyy" duct, the static pressure is zzz inWC.
OK, so now you have to figure out the duct sizes and airflow speeds. You know that you need to move 150 CFM of air through your room, but that is NOT a measure of speed! it is a measure of flow rate. Not the same thing at all. Think of water in a garden hose: you might have a volume of 1 liter per minute flowing out the open end of your hose quite slowly, but if you press your thumb part way across the end of the hose, the speed goes up a huge amount (you can now squirt water a long distance), even though the flow RATE is the same: it is still only 1 liter per minute, but you have increased the SPEED a lot by restricting the diameter of the flow: The same happens in your HVAC system. If the duct is narrow, then your 150 CFM must flow through it very fast, which makes a lot of noise. Slower is better. The rule for studios is that the flow speed has to be kept under 300 feet per minute, absolute max, and better still, under 100 fpm. (To give you some perspective, 300 fpm is roughly 3 MPH, and 100 fpm is roughly 1 MPH)
So now you know that you have to get a flow
volume of 150 CFM, and you have to allow it to move at a flow
velocity of no faster than 300 fpm. See where this is going? You can now figure out what duct size you need to achieve that goal! Bingo!
Simple math: What cross sectional AREA (square feet) will allow a VOLUME (cubic feet per minute) to move at a SPEED (feet per minute). High school math: 150 CFM / 300 fpm = 0.5 square feet. So the cross sectional area of your ducts needs to be 0.5 ft2 or more, which is 72 square inches (it's easier to work in in2, than ft2). Your existing brick duct measures 11 x 5, which is only 55 in2, so the air will be flowing faster than 300 fpm up there. Nothing you can do about that, since the duct is made of brick and you have no access to it anyway, but frankly I'm not concerned about how fast it goes through that duct, since it will be on the
other side of your silencer, and therefore any slight airflow noise in that duct won't make it back into the room, and it won't hassle your neighbors either, since it is just the sound of air moving, and will be very faint anyway. What I
am worried about, is duct sizes inside your room, and register sizes. You will need registers that are at least 10 x 8 inches (80 in2), and actually more than that, since the vanes in the register block off part of the total cross section. That's often expressed as "percent open area" of the register: So if your register has 70% open area, then you need registers with a
total area of 104 square inches, because only 72 of those are actually
open to the flow of air, and you need 72 square inches to get the maximum speed. In fact, you'd want registers that have a much larger area, in order to get the speed down as low as possible, because slow moving air makes less noise than fast moving air. A 12 x 12 register would be a good choice for you. And you also want 36" of perfectly straight 12" duct right before the register, to eliminate turbulent flow and the associated noise, before the register (the rule is that the straight run should be at least as long as three times the duct diameter). But that's just the final few feet of duct: What about all the rest?
All of the above will allow you to figure out how big those ducts need to be! You know you need at least 72 square inches of cross section in your duct, and if you use round duct then once again you can use simple high-school math to figure out the cross section: Area = PI x R ^2. But duct sizes are given in diameter, not radius, so divide the diameter by 2 to get the radius, square it, multiply by 3.141. An 8" duct has a 4" radius, and thus an area of 50 in2. Not enough. A 10" duct has a radius of 5" and an area of nearly 79 in2. Perfect! So you need 10" diameter ducts (5" radius).
Finally, with all the above info, you can design your silencer box. Here are some examples of what silencer boxes look like:
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Basically, just a large box made of thick wood, with an entry port on one end, an exit port on the other, several baffle plates in the middle, and the entire interior lined with 1" duct liner.
Great! ... but how BIG to make it? Once again, math and science comes to the rescue:
Silencers work on several principles at once. The first principle is that when sound waves moving through air experience a sudden change in acosutic impedance, part of the wave bounces back, and the other part loses energy. That is called "silencing"! Good stuff! In a duct, you can create a sudden change in acoustic impedance by suddenly changing the cross sectional area of the duct. It turns out that changing it by a factor of 2 is a good thing. So you want to make the box suddenly have twice the area where the duct comes in, then suddenly go back to half the area where it goes out. Cool! So at the point where the 79 in2 duct comes in, then box needs to have an internal free cross sectional area of at least 158 in2. Call it 160, to make the math easy. So it has to measure at least 16" x 10" internally. But wait! I said you need that much OPEN space, but that you also need an inch of duct liner on each side of the box, inside. So the internal size of the wooden box itself needs to be art least 18 x 12, in order that the open area left after you add an inch of duct liner on
each side, is 16 x 10. That's the MINIMUM size you need: you could easily make it 20 x 14 on the inside if you wanted (and can afford the space in your room), to get greater impedance mismatch, but even at 18 x 12 you'd be good.
Ok, so that's the cross sectional size of the box... but how LONG do you need to make it? That's determined by how many baffles you want to put in it. Since the baffles need to be far enough apart to keep the same cross sectional area, and they are also covered with an inch of duct liner, in your case you'd have to have them either 18" apart (if the box is 12" deep) or 12" apart (if the box is 18" deep). Your choice. So if you want just two baffles in there spaced at 18", then your box would have to be 3x18 = 54" long (why "3x" for only two baffles? Because 2 baffles splits the box into three parts...). If you wanted three baffles it would have to be 4x18 = 72" long. 4 baffles would make it 5x18 = 90" long.
So how do you figure out how many baffles you need? That depends on how much acoustic isolation you need. Which is why you need to do the test that FriFlo suggestied; meaure levels with a sound level meter, so you can figur out how loud you are (in dB) and how quite you need to be (in dB). More isolation = more baffles (and thicker wood). You need at least one baffle, and two is recommendable. (I have been known to put as many as five in a huge box made from two layers of 3/4" plywood, to get very high levels of isolation....)
Now for the fun part! If you put in more baffles, or if you have narrow ducts, then your static pressure goes up for the entire system, meaning that the flow rate drops, the air velocity goes up (thus making more noise), and the fan has to work harder (thus making more noise). So you need to re-calculate all of the above!
In other words, it's an iterative process: You need to go over it a few times, tweaking numbers here and there, until it all makes sense: The limiting factors are the air speed (never faster than 300 fpm at the register, and preferably more like 100 fpm), and the static pressure (most fans can't handle more than about 0.5 inWC of static pressure). If need be, you could even drop the flow rate (CFM) down a bit below the recommended 6 room changes per hour, so you can get smaller ducts, smaller silencer boxes... but higher static pressure!
(HVAC is fun, isn't it!)
So that's how you do it: as FriFlo said: there is no simple formula for this: there's a whole bunch of them, and they all depend on each other, and interact with each other, and it is up to you to fiddle and tweak until you get it right.
But ball-park: You'll need a fan that can move 150 CFM or air, 10" ducts, 100 square inch registers, and a silencer box that measures 18 x 12 on the inside and is at least 52" long. The box would be made of at least 5/8" plywood or MDF. If you want more isolation then make it with 3/4" wood and make it 72" long with 3 baffles in it. If you want fantastic isolation, then make it 90" long with 4" baffles in it, and make if from two layers of 5/8" plywood.
Of course, you need two boxes like that: one on the existing duct (which I'll assume is the exhaust duct) and the other on the intake duct (which doesn't exist yet). The fan should go on the exterior inlet port for the supply duct, so it won't be heard.
Where to put the box? Wall or ceiling? Wherever it will fit best! Yes it will change the room acoustics slightly, but probably not enough to worry about. If this is a control room, then put the box someplace where it does not mess up the left-right symmetry of the room. If it is a live room, then that's not an issue. In your case, you are a bit limited by the fact that one end of the box has to mate with that huge hole in your wall, but from that point on the box can go in any direction: down the wall, across the top of the wall to the left, across the top of the wall to the right, or across the ceiling.
And when all that is in place, you then need to do the math to figure out the sensible heat load of your room, and the latent heat load for your room, so you can buy the right size mini-split system ito take care of your humidity and temperature issues. But that's a totally different subject, not related to your question about the vent....
- Stuart -
Basically, the ambient noise level of their room is higher than anything leaving my room. So, I just need to build the mufflers and put them into place... Until then, I'm stuck with a slightly stuffy room without proper ventilation. There are worse things. I think I'll try to get the build together next week or the week after. I'll post pictures of the process!