# Find Formula to Heat Water to Constant Temp | Hydraulic Systems Designer

• deckart
In summary, the hydraulic systems designer wants to heat water to a specific temperature by pressurizing it and forcing it through a relief valve. He is limited in his math skills, and is looking for a formula to calculate the relief setting needed to maintain a particular temperature. He is also looking for someone to help him do this.
deckart
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?

I want to heat water to a specific temperature by pressurizing it and forcing it through a relief valve.
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.

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.

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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 until it's warm, then take it out of the system all at once and replace with cold water.

Thank you, Q. This gets me a lot 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.

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.

Q_Goest said:
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 until it's warm, then take it out of the system all at once and replace with cold water.
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.

Q_Goest said:
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.

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.

Q_Goest said:
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 until it's warm, then take it out of the system all at once and replace with cold water.
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.

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.
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.

Astronuc said:
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.

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.

Astronuc said:
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.

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.

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?

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.

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.

deckart said:
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.
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 ??

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.

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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?

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.

Artman said:
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.

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
(15gpmX1800psi)/1714=15.78HP

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 a lot 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.

deckart said:
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.
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.

Consider the first law. The amount of energy that goes into the water is a function of the work put into it. If it takes 15 hp of work to compress the water, then 15 hp of energy has gone into the water. That converts to 15 hp of heat if the fluid is isenthalpically throttled back to the initial pressure.

If its hard to understand this, think about putting the entire system into a control volume. Put your control surface around the entire system which contains the pump and the water circuit. Now allow only the work input from the pump to cross the boundry. It doesn't matter how efficient or inefficient your pump is, all the work that crosses the control surface goes into the water in the form of heat as long as the water in the entire loop is within the control volume. Note also this assumes the entire system is insulated such that no heat is lost through heat transfer to your surroundings.

If you wanted to, you could have a small water tank at low pressure. The tank would supply water to a pump which is fed into a valve (relief valve, back pressure regulator, etc...) and the valve feeds the water right back into the small tank. The entire system would be insulated to prevent loss of heat. On the outlet of your valve, you'd also have a "take off" line to remove some water when it was hot enough. (You could also take the water off upstream of the valve if you wanted it at higher pressure.) You'd measure this with a thermocouple and control outlet flow with a control valve. You'd also have a "make up" line going into the tank which allowed fresh, cold water into the tank at the same rate you removed it from your take off line.

The equation for this is what I'd given above, with the mass flow rate being the flow coming out of the take off line.

Yes, to heat the water I'm using a positive displacement high pressure pump (displacement efficiency is over 95%) and moving the water directly over a relief valve @ 1800 psi then recirculating it until 120 F is reached. Then it will be available for use. No work is being performed. Head pressure alone will compensate for the water being moved.

Originally I was going to vary the pressure proportionally but I've realized that this is not necessary.

I see, I was thinking centrifugal with the motor outside of the system.

High pressure water hydraulics (aside from jet cutters) is a relatively new industry. I don't believe any high pressure water hydraulic components are even made in the US. Europe, having higher environmental standards, is where it is primarily being used.

http://nessie.danfoss.com/Products/index.asp

BTW, Artman, thank you for the formulas.

## 1. How do you determine the formula for heating water to a constant temperature?

The formula for heating water to a constant temperature can be determined by using the specific heat capacity of water, the initial temperature of the water, the desired constant temperature, and the amount of water being heated. This formula is known as the heat equation: Q = mcΔT, where Q is the heat energy required, m is the mass of water, c is the specific heat capacity of water, and ΔT is the change in temperature from the initial temperature to the desired constant temperature.

## 2. What factors affect the heating of water in hydraulic systems?

Several factors can affect the heating of water in hydraulic systems, such as the flow rate of the water, the pressure of the water, the size of the piping, and the efficiency of the heating system. The type and size of the heating element and the insulation of the system can also play a role in the heating process.

## 3. How does the heating process in hydraulic systems impact the overall efficiency?

The heating process in hydraulic systems can have a significant impact on the overall efficiency of the system. If the water is not heated to the desired constant temperature, it can affect the performance of the entire system. Additionally, if the heating system is not efficient, it can lead to higher energy costs and potentially damage to the system.

## 4. Are there any safety considerations when designing a hydraulic heating system?

Yes, safety should always be a top consideration when designing a hydraulic heating system. It is important to ensure that the system is properly insulated to prevent any potential hazards from hot water. Additionally, proper maintenance and regular inspections should be conducted to ensure the safety and functionality of the system.

## 5. How can the formula for heating water to a constant temperature be applied in real-world applications?

The formula for heating water to a constant temperature can be applied in various real-world applications, such as in heating systems for homes and buildings, industrial processes, and agricultural irrigation systems. It is also commonly used in the design of hydraulic systems for machinery and equipment that require heated water for their operation.

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