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I need a formula

  1. Sep 2, 2005 #1


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    This won't be easy. I'm a hydraulic systems designer but my math skills are limited to basic algebra which would make it a challenge for someone to give me a formula to do what I'm looking for. Here goes:

    I want to heat water to a specific temperature by pressurizing it and forcing it through a relief valve. The problem being the flow will vary but I need the final temperature to remain constant. I can proportionally regulate the relief setting (Delta-P) as the flow changes. So, I need to know what the relief setting needs to be at particular flows to maintain a particular temperature.

    Any takers? :smile:
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  3. Sep 2, 2005 #2


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    From that I'm assuming you will pump the water to pressurize it followed by a throttling to lower pressure. Is that correct? You may want to be more specific, especially WHY do you want to do it this way? Just a side note, a relief valve might not be the best solution unless the valve can throttle properly like a back pressure relief valve. Some relief valves have problems doing this as a regular operation.

    I'd assume you have water at some initial pressure from which you need to pump it, and then expand it. Is that correct? Is the initial temperature constant? How much does the temperature need to increase, from what to what temp?

    By adding 'work' to the water, you can increase the temperature, but because water is so incompressible, you won't add much energy by pumping it, and therefore the increase in temperature will be small. You could reintroduce the water to the pump and keep cycling it, and that would eventually increase the temperature but that would not be very controllable without temperature sensors and proper PLC controls. The water would only warm gradually after each cycle through the pump and valve.

    In principal, its doable. Assuming you insulate everything to keep from loosing heat, the water is isentropically compressed followed by an isenthalpic expansion. Putting together the equations is fairly simple, but the problem could use better definition first.
  4. Sep 2, 2005 #3


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    I acknowledge that this is not the most practical way to heat water. The most efficient way to heat it would be to put it in direct contact with a electrical heating element. But, none-the-less, suppose commercial electrical energy is not available.

    The relief valve would be the most direct way to regulate the delta-P. The water would be moved with a variable displacement pump as needed driven from the PTO shaft of a truck or other mobile equiptment. Transducers would monitor water temperature and software would run a driver card that would in turn proportionally drive the relief setting as required. (electrical components powered by 12V or 24V mobile battery) The water would be cycled until the temperature is reached and then released as fresh cold water is introduced. The temperature of the water and flow are variables. The constant would be the final water temperature.
    Last edited: Sep 2, 2005
  5. Sep 2, 2005 #4


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    Ok, great! Having controls on this system and circulating makes this doable. The equations get much simpler also, since you can simply draw a control volume around the entire system and say water energy (ie: enthalpy) in plus energy in (in the form of pump then pressure let down) equals water energy out. You don't need to look at each individual pressure increase and decrease.

    PP = m * cp * dT
    Where PP = Pump power (energy going into the pump times time)
    m = mass flow rate
    cp = heat capacity of the water
    dT = temperature increase

    This gives you a rate at which the water will increase in temperature and the length of time. Note that the mass flow rate is NOT the pump flow rate though, it is the mass flow rate into and out of this system assuming the flow is constant. You may want to solve for the mass flow rate you want out of your system, then size the pump accordingly. Note also it doesn't matter how much pressure you're going to, only power consumed by the pump. Obviously a higher pressure pump will have a lower flow rate when compared to a lower pressure pump but that's not important. What's important is how much energy you're putting into the water with your pump.

    Since you're doing a batch job, your mass flow rate isn't constant, but for the sake of arguement, assume it is. All you need to do is keep recirulating water untill it's warm, then take it out of the system all at once and replace with cold water.
  6. Sep 2, 2005 #5


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    Thank you, Q. This gets me alot closer to what I'm looking for.

    An application might be a military field shower tent (to be used in New Orleans right now for ex.) to where the water could be heated as needed on site.
  7. Sep 2, 2005 #6


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    Hmmm..... why not just an electric or other type of heater powered by propane or other fuel with a very small pump if you need pressure? Seems to me if you have electric power for a pump, you have power for an electric heater.
  8. Sep 2, 2005 #7


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    I was thinking the same thing. We do the same here on our reliefs, but with a large coil to dissipate the heat. We relieve back to the inlet of the pump. Since it's a variable pump, relive the flow back to the inlet and make that a sizeable volume. Once that heats up you can then bleed off that to the demand side or slow down the recirc on the pump relief loop. You may have to play with pump speeds to control the temperature, but that definitely sounds like a quick and easy way to go about it.
  9. Sep 2, 2005 #8


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    At this point, I'm just looking at different ways to accomplish the same thing. As a prime mover to drive the pump, I was talking about the PTO shaft of an engine on a truck or tractor. Where a generator or commercial electricity is not available.
  10. Sep 3, 2005 #9


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    While there is a water supply (batch) the pressurs could be held constant with a reciprocating pump.

    I believe such systems are used in hydro-saws (or hydrulic cutting systems), where extremely high pressures are used to produce high velocity water jets to cut through hard substances. The water is collected and recycled.

    I believe also the nozzles are made of sapphire or similarly hard substance due to the erosion of softer materials.

    But if one wishes to shower, one would have to allow the water jet so expand.

    I think heating units would be more practical.

    Heating from pumps is used in nuclear reactors before the core power is increased. On a typical 1100 MWe unit (PWR), each primary recirculation pump provides about 6-8 MW of power, and the initial heat up is done by running the pumps.
  11. Sep 6, 2005 #10


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    Ah, interesting.

    In mobile hydraulics, too much heat in the system is a common problem. So it got me thinking of how you could intensionally create heat and it actaully be useful for something.

    I'm aware that there are electric heaters for water that only heat the water as needed rather than heating an insulated reservoir. There are probably fossil fuel versions of the same thing. But if all you had was diesel rig out in the bush and a reservoir of fresh water, what are some other ways you heat it immediately as needed? I have just been exploring this as an alternative. If the unit could be small enough/portable and heat water on demand it might have a few useful field applications when other energy sources are not available.

    Thank you for the input.
  12. Sep 6, 2005 #11


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    Actually, water cutting uses abrasive water jets: The water contains abrasives that do the cutting proper - which is also why the nozzle needs to be hard. It would take some pretty serious water velocity/pressure to cut through inches of steel without that.
  13. Sep 9, 2005 #12
    Can this work? Water being a liquid and largely incompressible won't react quickly to variations in pressure to vary temperature, unlike a gas such as steam. What temperature is the water starting? Or am I misunderstanding the procedure? How much difference are you looking to achieve and what kind of flow rates are we talking about?
  14. Sep 9, 2005 #13
    There weren't any posts on this when I posted. I guess the others were written in the double post and dropped in here. Okay, friction of passing the pressurized water through a restriction could produce heat.
  15. Sep 10, 2005 #14
    Water is not a gas!!
    In fact when you compress water hard enough it will reach the temperature of 4 degrees celcius (the temperature at the bottom of the ocean) this is the temperature when water has its highest density.
  16. Sep 14, 2005 #15
    no one ( i think ) has mentioned this, if you had access to a machine shop , you could create an exhaust manifold that circulated the exhaust gasses through a heat exchanger , heating your water ..
    if it is a large diesel engine you could use the cooling water itself !!and use the thermastat of the engine to control the flow of hot water..
    on the practical side ..there is a lot of wasted heat from a diesel engine..
    you could take out the radiator , but you would need to monitor the temp of the cooling water..
    What size diesel are we talking about ??
  17. Sep 14, 2005 #16


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    Well, you would have to modifiy a particular machine to heat the water with the engine directly. With this pump/heater idea it would be a portable unit you could simply run of the PTO shaft of a variety of machines. Now there are HP limits on a typical PTO. I don't know for sure but maybe 10-15 HP? And that alone would limit the amount of water you could heat at a given time. But how much that is, I really don't know and that's part of what I'm after.

    The idea is to have something versatile across a number of machines with a minimal amount of modification and/or parts need to mount on the prime mover. And that it could sufficiently heat water using the rotary power of the engine. It would be a relatively clean heating unit that could be used on whatever heavy vehicle happens to be in the area where the hot water is needed. I could see this being used in variety of field applications in remote areas.
    Last edited: Sep 14, 2005
  18. Sep 14, 2005 #17


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    We are having more discussion of this at work...

    Trying to get closer to a formula. To break this down one way (out of several) we can look at it like this:

    We have a set amount of HP availabel - 15 HP
    We have 30 Gallons of water
    We want to raise the temperature from 40 deg F to 120 deg F.
    How long would it take?
  19. Sep 14, 2005 #18
    Okay, I think I have enough to give you an idea of the formulas required.

    First we need to find the heat lost from the pump.

    Every HP uses 746 watts/hr. Assuming 60% efficency, 60% goes to running the pump, the other 40% is your heat.

    Watts of heat = 746w/hr x 15hp * .40 = 4476w/hr

    I then would convert this to Btu because I am used to this and it's an easy jump then to your answer

    4.476 Kw/h * 3.412 = 15,270 btuh

    Now we need to know the weight of the water:

    8.33 lbs/gallon * 30 gallons = 249.9 lbs

    1 btu will raise 1 lb of water 1 degree. So 15270 btuh / 249.9 = 61.1 deg/hr

    120 deg F - 40 deg F = 80 deg F

    80 deg F / 61.1 deg/hr = approximately 1.3 hrs

    Remember, this assuming that you can capture all of the watts of heat lost to inefficiency (which you can't). So it should take longer.
  20. Sep 15, 2005 #19


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    Awesome! That begins to give me a feel of how this might work.
    Ok, thinking out loud:

    1.3hrs(78min)/30 gallons = 1 gallon heated to 120 deg F every 2.6min

    But I'm thinking it will work better than that in this situation.

    Now all this HP is directly converted to heat because none of it is doing work. Other than heat absorbed by the pump valve assembly itself initially, it should all go directly into the water. So the efficiency to actually heat the water might be 95+ %, wouldn't you agree? Most of that 60% efficiency lost by the pump should go directly to heating the water, I would think.

    Unless I'm mistaken:

    Watts of heat = 746w/hr x 15hp * .90 (as apposed to .40) = 10071w/hr

    10.071 Kw/h * 3.412 = 34,362 btuh

    34,362 btuh / 249.9 = 137.5 deg/hr

    80 deg F / 137.5 deg/hr = approximately .58 hrs (34.9 min)

    .58 hrs(34.9min)/30 gallons = 1 gallon heated to 120 deg F every 1.16min

    At this rate you wouldn't heat alot of water as fast as I would like. But this is enough to convince me to put together a prototype and test it.

    One of the enemies in this scenerio is the lower ambient temperature of the atmosphere where the heat will tend to dissapate. The unit will be insulated to some extent so as to retain the generated heat and keep it in the water.
  21. Sep 15, 2005 #20
    I'm not so sure. Is the water being moved? That is work being done and the estimated 60% would go to that. Unless you are intentionally selecting a poor efficiency pump, which you could do, and increase the heat lost to inefficiency.

    However, resistance heat in the line would work better it's 100% efficient and some sort of vapor cycle such as a heat pump discharging heat to the water can even exceed 100% as it is an overunity device.
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