what is the effect of mixing 1" pipe, 3/4" pipe and 1/2" pipe

You should stick to the flows for each pipe size I mentioned above, They are based on the industry standard that is probably80 years old.

If you install a larger circ pump to get more flow through smaller pipe you will get velocity noise and possible but not likely erosion of the pipe or tubing.

The proper steps to follow in the right order are:

1. Heat loss of the room
2. size of radiation to meet the above heat loss and calculate water flow
3. size pipe based on water flow
4. size circ pump based on water flow and pipe size.

If you want it to work without issue you have to follow the steps in that order above and one thing builds off the the other

If you jump into the middle or try to work backwards you will have issues. Guessing does not work.

Hgher flow rate thru te 1/2" pipe may get you from 1.5 GPM to 2 GPM which will give you about 5,000 BTU more in that loop with 1/2" pipe.

The better design will be 3/4" or larger for all the pipes. So I believe your bathroom radiator has 1/2" pipes going to the basement, the kitchen and living room has 3/4" going to the basement.

A more powerful pump may be enough but it will make the 1/2" pipe noisy. The reason to keep the flow thru 1/2" pipe at a maximum of 1.5 GPM is to reduce the noise.

Think of it like this. there is a 1/2" pipe feeding a 3/8" tube that fills the toilet tank. That has 45 to 70 PSI city water pressure behind that closed valve. when you flush, you hear the flush and when that noise goes away, you still here the water filling the tank until the tank is full and the float valve is closed. Now that noise in the pipe is normal and you do not think twice about it because you know that noisy pipe will stop in less than a minute.

Now think of that same noise in the 1/2" pipe every time your thermostat calls for heat. That noise continues for 20 minutes until the thermostat is satisfied. That will bother some folks late at night while in bed with no other ambient noise except the water running thru that radiator pipe. So you may not want that 1/2" pipe in the series loop system.

So, the flow will be like 1" pipe → 1/2" -> 3/4" (baseboard) → 1/2" → 1" pipe for the longest run .

I would leave that radiator out of the loop and make it a 1/2" or 3/4" PEX home run like the boiler room.

This may have been designed by someone who really knew what they were doing. if the sketch is accurate, it looks like we have zones of series loop heating with returns back to the boiler. They may have relied upon laminar and turbulant flow rates to keep even heating throughout the home.

Your original issue is that some of the rooms are warmer than others and other rooms are cooler. You described that the home is not balanced. Each time you add a little more detail.

The best info so far is the floor plan. I took an educated guess at the location of the boiler in the basement and the radiators in the rooms. Based on this diagram, can you tell me if this is an accurate illustration of the design?

As you can see, I have incorporated as much info from your different comments. The Dark Red illustrates the Baseboard radiators, and the one cast iron radiator in the front entry on the first floor, The faded red illustrates the basement pipes and the 45° angle pipes illustrate the the risers from the basement to the second floor.

If this illustration is close to accurate, then it would also be helpful to know what rooms are cold in comparison to the rooms that are hot. If you can list the rooms from coldest to hottest that would be good in determining how to solve the problem.

I have made assumptions based on how the system is designed that have been proven incorrect. So I may have offered solutions that may be more involved that needed. This may not be as hard as I originally stated. Another helpful fact would be if the second floor system was ever balanced in years past. From one of you diagrams it appears that the smaller bedroom and the Den/Office radiators are connected together end to end without dropping the the basement. Your diagrams also indicate that there are 2 separate MonoFlo® loops that split off of a 1-1/4" main and rejoin at the other end of the basement to connect to a 1-1/4" return main. If this is also correct, there may be a balancing problem in that design. You may only need to do a littler basement pipe redesigning to solve your balance problem.

I will not recommend a solution until I have answers to

1. Is the basement 1" loop as I have described: actually 2 different loops?
2. What rooms are cold, what rooms are hot?
3. Has this been a problem since the building was built, or is this something that has happened in the past several years?
4. If question 3's answer is "Has not always been this way", then when did the problem start to happen?

Knowing the answers to these questions will help me make the next recommendation.

Great information. This helps. The ONIX tubing is not original to the age of the building. Therein lies the problem. A repair was made at some point before you owned the building. When the 1/2" tubing was introduced to a well balanced system the balance was thrown out of wack. So using any of that 1/2" ONIX tubing is going to to cause you problems. All those 1/2" tubes will need to be replaces with 3/4" tubing. I think that you don't need to redesign the system, all you need to do is replace those ONIX tubes with 3/4" PEX with Oxy Barrier tubing.

Is this diagram a little closer to your existing system?

Now to address the thermostat setting when the first floor is unoccupied, (you need to keep the pipes from freezing so you should Leave the heat on at at least 50°F. As the first floor heat, when occupied, will make the floor of the second floor apartment warmer, there is less heat needed to heat the second floor when first floor is occupied than when it is unoccupied.

When both apartments are occupied the first floor does not have a cold ceiling, so it will always use less fuel that the second floor. The second floor will always use more fuel because it will loose heat thru the ceiling to an unconditioned attic, or if you have a flat room directly outdoors.

The Monoflo® one pipe design can be unbalanced at the point where the 1-1/4" splits into two 1" loops. If after you get all the 1/2" tubes replaced with 3/4" tubes you my find that the bedroom side gets more heat than the Living room side. You can fix this with balancing valves placed on each of the 1" branch pipes. as shown in this diagram.

And one note: did I get the bathroom to Kitchen radiator connection correct? From one of your pencil on graph paper diagrams I thought that the bathroom had 2 basement runs and the kitchen had 2 basement runs. I changed it to one 1/2" Onix run from the basement to the bathroom, then an original 3/4" copper run from the bathroom to the Kitchen, then a 3/4" copper basement riser from the kitchen to the basement.

This 1/2" to 3/4" monoflo® branch is a reason for improper balance. That is why I suggest that you find a way to replace all the 1/2" Onix with 3/4" PEX.

If there is no access to hide the 3/4" PEX in the first floor apartment wall, I might suggest that you run the PEX Exposed then box it in with pine boards and paint it the color of the wall

Also you may want to have the first floor heater share one of the hallway radiators in some fashion. It is not fair to have the second floor tenant foot the bill for all the common area heating. especially since that gas bill will always be higher on the second floor.

@JR3_Home_Performance

It is interesting that the first floor may have a larger load. But the heat loss calculations that everyone uses does not account for that. The Manual J and my Hydronics Institute forms account for infiltration equally from floor to floor. That makes the math add up to more outdoor unconditioned surface area on the second floor with a ceiling that is exposed to the outdoor temps while the first floor ceiling is exposed to a heated room above. But that is just the math not the fact that some of the second floor's infiltration is from the first floor's heated air. As I said… Interesting

The OP has been trying to find the answer to the unbalanced system for some time over 4 different discussions herein. I have followed the comments on all of them and I am learning a little bit more each time. The OP has asked plumbing professionals to solve the unbalanced system problem and has received different Quotes that are in some cases contrary to common sense.

I might suggest that whoever gets the job should include a performance guarantee. Perhaps 1/2 down and the other half when the system is performing satisfactorily next winter when the temperature drops below 30° for 3 days in a row.  That is the only way to get someone to get it right.  

There was a saying “You Got To Sweat for your Money”. 
That came from an old boiler installer requirement.  When installing a boiler and radiator system in the middle of the summer,  the contractor needed to jump the thermostat on the burner or fire the coal boiler to maximum on that hot summer day.  The inside temperature needed to get to 130° from an outdoor 80° evening temperature.  This ensured that the boiler could handle the load on a cold winter day.  Lots of candles melted from those tests but that is what was needed if the boiler installer wanted to get paid. 

You can follow the OP's other discussions to get the whole picture. You can see where I was offering a completely different solution with the original information received in this discussion. Be sure to click on page 1 of the discussion to get to the beginning of the post.

Here are the other two

Okay I see it was 60kbtu so that size expansion tank should be okay but it wouldn't hurt to remove that one radiator to reduce water volume unless that stairway really needs heating

OK now I believe I have all the needed information.  (I Hope)

This diagram will show what I might propose to solve your situation.  

1. Cut and remove or abandon the green pipes in the illustration.  Place 90° elbows where the Tee fittings used to be.
2. Remove all of the Monoflo tee fittings
3. Connect the living room to the bathroom in some way, this will involve the new ¾” blue pipe (or PEX) in the illustration.  Try to snake the ¾” PEX thru the wall where I have drawn a brown square.  I believe that is the chase where the chimney goes from the basement to the roof.  
4. If you can not see a way thru that area, then you need to find the chase where the drain and vent pipe for the toilet is located and try to snake the PEX thru there.  If that is unsuccessful, then run the PEX pipe exposed in the first floor bathroom.  Preferably near the corner.  Then you can cover that exposed pipe with a piece of molding and paint it the color of the wall.  
5. This will complete the series loop for the living room side of the apartment.  

On the bedroom side

1. Connect the front bedroom to the middle bedroom as you indicated.
2. Use the good ¾” copper riser as the return to the basement and connect that to the other end of the front bedroom radiator with a ¾” PEX that is inside the baseboard cover.  Use the U-Turn fitting or 2 slip 90° elbows to connect the PEX to the far end of the radiator.
3. This will complete the series loop for the bedroom side.  

Your existing circulator will be able to handle that much piping friction loss for both zones.

You can use balancing valves on the 1” pipe or you can use zone valves with separate thermostats for the Bedroom side and the Living room side.  

As far as the Cast Iron Radiator, I would see how easy it is to add it to the first floor boiler using Monoflo® tees on the series loop and try to place it on the return if possible.

I would offer you this if I were not retired and offer you the performance guarantee also.  I am confident that this plan will work.  The glitch is getting the pipe from the second floor to the basement.  Exposed piping is often the best and only option.  the file below is the Diagram above with the floor plan overlayed.

"This is why I proposed the 2 series loop solution since two separate loops make each loop shorter . And each loop drives only half the load. I hope this makes sense."

Good design. I would use zone valves in your situation.

"Also, what is this new ¾” blue pipe (or PEX)? I would love a link on this blue pipe. For the 3/4" PEX for heating, any recommendations for proven choice of PEX? I am mostly afraid of freezing given the basement is not insulated (about 1/3 to 1/2 above ground)"

There are several PEX tubing Manufacturers, Just select any oxygen barrier tubing for a closed system here is one that will work just fine. I have always used PEX-a The Wirsbo but I believe the PEX-b is just fine.

https://www.supplyhouse.com/Bluefin-T075-300-OXY-3-4-Oxygen-Barrier-PEX-b-Tubing-300-ft-Coil

https://www.supplyhouse.com/Bluefin-T075-300-OXY-3-4-Oxygen-Barrier-PEX-b-Tubing-300-ft-Coil

I believe that your Black Onix tubing was the replacement for frozen pipes. So placing the pipes inside the conditioned space is important.

"The chimney you drew was there, but has been fully closed in both unit1 and unit 2. I think the major challenge is to snake the pipe through living room to the bathroom, and from bathroom to the kitchen without going down to the basement."

It will be necessary to open the floor in the bathroom or the ceiling in the first floor apartment to access the bathroom radiator piping and get rid of the 1/2" risers. If you open the ceiling below, you may be able to make that series loop happen without going to the basement. If the bathroom has a tile floor that you may not want to disturb, then using the ceiling of the second floor bathroom may give all the access you need to snake PEX tubing everywhere you need to go. from the living room to the bathroom and from the bathroom to the kitchen. Are there soffits above the first floor kitchen cabinets?

Based on the BTU output of the zone, the 3/4" PEX is just fine. See the problem you have with your 1/2" Tubing is the actual ID of the rubber Onix is smaller than the ID of 1/2" Copper, hence the additional restriction.

temperature changes cause pressure fluctuations

To give you an example of how pressure might change:

Say you have a 100 gallon tank of water at 60°F.

If you heat up that 100 gallons of water to 210°F the actual volume of that same water will expand to about 104 Gallons.

If your container is completely sealed and can only hold 105 gallons of water, then when that water expands there will only be 1 gallon of compressed air in the top of that pressurized tank. That one gallon area of air may increase in pressure from 0 PSI to 20 PSI when that happens.

If the tank only holds 103 gallons of water then there will be no room left in the tank for air, so the Hydrostatic pressure can build up to over 1500 PSI and eventually the tank will explode.  That is why boilers have relief valves.

If however, you have a 120 gallon tank for that 100 gallons of water, then when the water expands to 104 gallons there is still 16 gallons of air space in the tank, so the tank pressure will only go from 0  to about 5 PSI since there is lots of room left to compress the air.

These numbers are not mathematically accurate but are for illustration purposes

Now put that same logic in your boiler with all the connected pipes and radiators.  That is a sealed system.  And when it was installed there were two important parts that were included.  The relief valve, just incase the pressure from the expanding water gets above 30 PSI like the the 103 gallon tank example.  The other is the expansion tank.  That is the place where the expanding water goes.  If you have an overloaded or flooded expansion than (that is a tank with very little air space leaf because it is loaded with too much water) then there's no room left for expansion and the water pressure in the boiler will increase more than it would if the tank had the correct amount of air in it.  

Some expansion tanks are just tanks. The air will compress as you fill the system with 12 PSI water.  Those tanks have both air and water in them with nothing separating the air from the water.  The other expansion tank has a membrane or bladder that is pre-charged at the factory with 12 PSI air pressure; the membrane will separate the water from the pre-charged air side of the tank.   If that Air pressure is released over time, then the 12 PSI water will fill the tank and leave less room for expanding water.  When that happens, you will get a greater pressure rise in the boiler, similar to the 104 Gallon tank example.

I hope this will answer your question about how the pressure changes differently on identical boiler systems connected to different expansion tanks.  You probably had an overloaded expansion tank on the problem system. 

Mr. ED

Any luck?