# Temperature of fluid flowing through pipe

## Main Question or Discussion Point

Is there a mathematical relation for the temperature change when a fluid flows through a pipe from one end to another?

I am not aware of any equation in thermodynamics for this, but I would guess the following parameters are important :

1. cross-sectional diameter of pipe
2. viscosity
3. volumetric flow rate
4. specific heat of the fluid
5. length of pipe.

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Chestermiller
Mentor
Is there a mathematical relation for the temperature change when a fluid flows through a pipe from one end to another?

I am not aware of any equation in thermodynamics for this, but I would guess the following parameters are important :

1. cross-sectional diameter of pipe
2. viscosity
3. volumetric flow rate
4. specific heat of the fluid
5. length of pipe.
Yes, and you've identified all the parameters involved, except for the density and thermal conductivity of the fluid. See any book on transport processes or heat transfer. I highly recommend Transport Phenomena by Bird, Stewart, and Lightfoot.

This site has temperature analysis of pipe flow:

the pipe could be stationary or moving.

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Chestermiller
Mentor
The following site has a summary of what you are looking for for both laminar flow and turbulent flow in at tube in which the wall temperature is constant along the tube: http://web2.clarkson.edu/projects/subramanian/ch302/notes/Convective%20Heat%20Transfer%201.pdf [Broken]

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Andy Resnick
Is there a mathematical relation for the temperature change when a fluid flows through a pipe from one end to another?

I am not aware of any equation in thermodynamics for this, but I would guess the following parameters are important :

1. cross-sectional diameter of pipe
2. viscosity
3. volumetric flow rate
4. specific heat of the fluid
5. length of pipe.
Don't forget the temperature (and thermal properties) of the pipe itself.

Chestermiller

That pdf defines a number of constants for the fluid flow, but it doesn't give a more direct mathematical relation.

What I am trying to find is :

ΔTemperature = f(parameters)

Thank you.

---

Sunfire

Your link seems to say a frame of reference is important? The equation I need is for temperature from a exhaust pipe of a car (for instance), and the radiator hose.

The equations in the page seem to show that the temperature is only dependent on velocity and specific heat of the fluid. I imagine more factors are invovled.

Thank you.

----

Andy Resnick

Yes, the thermal properties of the pipe are important too. I am trying to get this equation for the exhaust pipe of a car and the radiator hose.

The equations in the page seem to show that the temperature is only dependent on velocity and specific heat of the fluid. I imagine more factors are invovled.
Hi,

you are most certainly right. The quoted page gives ΔT when the pipe velocity equals c and the flow exit absolute velocity is also c. This is the maximum cooling that can happen, assuming that both velocities match. This is the simplest case.

To fully understand how the entire phenomenon happens, you will need to approach it methodically.

1) You have to read about "Fanno flow" in a stationary pipe. Fanno flow analysis will tell you how the flow exit velocity depends on the friction factor and the length of the duct etc.
This is for motionless duct.

2) Once you know the exit flow velocity (from Fanno flow analysis), go back to the page and apply the temperature analysis with your particular parameters.

3) This will give you the ΔT you are looking for.

I hope this helped!

Chestermiller
Mentor
Chestermiller

That pdf defines a number of constants for the fluid flow, but it doesn't give a more direct mathematical relation.

What I am trying to find is :

ΔTemperature = f(parameters)

Thank you.

---

Sunfire

Your link seems to say a frame of reference is important? The equation I need is for temperature from a exhaust pipe of a car (for instance), and the radiator hose.

The equations in the page seem to show that the temperature is only dependent on velocity and specific heat of the fluid. I imagine more factors are invovled.

Thank you.

----

Andy Resnick

Yes, the thermal properties of the pipe are important too. I am trying to get this equation for the exhaust pipe of a car and the radiator hose.
If this is what you are trying to do, then the problem is much more complicated than just a simple explicit equation for the temperature change as a function of the parameters. Sorry.

For your situations, you are going to have heat transfer resistance within the pipe, heat transfer resistance through the pipe wall (as AR indicated), and heat transfer resistance on the outside of the pipe. In addition, for the exhaust pipe, radiative heat transfer (both inside- and outside the pipe) is probably going to be significant (and needs to be taken into account). The radiator is going to have cooling fins, and that is going to be important. You just need to learn heat transfer, or hire a heat transfer consultant. There is too much material to cover to present all this here. If you want complete coverage see Transport Phenomena by Bird, Stewart, and Lightfoot, or Heat Transmission by McAdams.

Chet

Do the books you mention describe the phenomenon in detail? Because I would only purchase them if they are useful in that aspect. Which one of the two books would you recommend?

Chestermiller
Mentor
Do the books you mention describe the phenomenon in detail? Because I would only purchase them if they are useful in that aspect. Which one of the two books would you recommend?
Yes. Both books do. Bird et al is a great book no matter what your motivation. it's a classic that was updated in ~2000 with the 2nd Edition. This is the one book that I used during my 40 year professional career more than all the others combined.

Chet

Thanks Chestermiller, I got the book. Could you tell me which chapters are relevant for me to find the equation I am looking for?

Chestermiller
Mentor
Thanks Chestermiller, I got the book. Could you tell me which chapters are relevant for me to find the equation I am looking for?
Chapters 10 - 16, particularly Chapter 14 and possibly chapter 16 (if radiation transport is important).

Hi,

I am not sure which equation actually refers to this. In any case, I am actually obtaining the temperature as data using sensors in my project.

But I still need another equation :rofl:

The equation for either the Power or Energy given off by such a fluid when its temperature has risen by ΔT.

My immediate guess is Q = mcΔT which I learnt in school. But would it be correct to use this equation for the radiator hose and exhaust, which have fluids in motion?

Thanks.

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I found the equation I was looking for. Its 14.6-1 to 3 in the book. I might be coming back here for some help though again. Thanks

Hey Chestermiller,

Could you tell me why would the equation for temperature difference be related to energy and not power (energy per unit time)?

In my case here, I have a fluid that is constantly being heated up (by the engine), so would there be a corresponding equation for heat POWER instead of heat ENERGY?

Chestermiller
Mentor
Hey Chestermiller,

Could you tell me why would the equation for temperature difference be related to energy and not power (energy per unit time)?

In my case here, I have a fluid that is constantly being heated up (by the engine), so would there be a corresponding equation for heat POWER instead of heat ENERGY?
The engine power is only a fraction of the rate of energy from burning the fuel. Much of that energy goes out the exhaust, and, if you don't cool the engine by the radiator, you will be very unhappy. The amount of heat removed by the radiator is not equal to the engine power.

Rate of Energy released by burning fuel = (engine power)+ (rate of heat out exhaust)+(rate of heat out radiator)....roughtly

Thats right.

What I am doing is measuring the temperature of the heated fluid coming out of the exhaust and through the radiator. But temperature is not the same as heat, so thats why the search for the equation that correlates temperature to rate of heat energy through these (heat power).

This is far off what I have studied in physics. Thank you for helping me. I am still reading Bird's book, didn't find anything on this though.

Hi Chestermiller,

Two equations for rate of heat energy are present in the pdf attached. In the topic IV, Heat Exchanger Subsystem do equations (2) and (3) give us the same quantity expressed as a function of different constants?

I would be using equation (3) but is the Qa for equation (2) a different one, or the same?

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Chestermiller
Mentor
Hi Chestermiller,

Two equations for rate of heat energy are present in the pdf attached. In the topic IV, Heat Exchanger Subsystem do equations (2) and (3) give us the same quantity expressed as a function of different constants?

I would be using equation (3) but is the Qa for equation (2) a different one, or the same?

I feel like, if your goal is to model the entire fuel system and cooling system, you may have underestimated the complexity of the problem, and you are going to have to develop a lot of fundamental background before you can do this.

Chet

1 person
Sorry for the late reply Chestermiller.

Right now, I want to just deal with the energy wasted through the exhaust (not the radiator).

The temperature sensor I have can only detect the heat at the outside of the pipe. I don't have a sensor that can be put inside the starting of the exhaust. Due to that I was hoping to make use of equation (3) { Qa = hAΔT } to make an estimation of the energy.

What I had in mind was placing one temperature sensor at the beginning and another at the end (both outside the pipe) to get a temperature measurements. And then using those to calculate the rate of heat energy.

So whats the best way to do this given the type of sensors that can't come in direct contact with the gas itself?

Chestermiller
Mentor
Hi Chestermiller,

Two equations for rate of heat energy are present in the pdf attached. In the topic IV, Heat Exchanger Subsystem do equations (2) and (3) give us the same quantity expressed as a function of different constants?
The two equations must match up when you eliminate the Qa's.
I would be using equation (3) but is the Qa for equation (2) a different one, or the same?
It's supposed to be the same.

Chet

Chestermiller, I really really appreciate your help. The thing is I am not able to communicate effectively with the professor who is supposed to be helping me through this. At the same time, its pretty late to be changing what I am doing and he always seems busy to respond.

1. If Qa is the same in equation (2) and (3) I assume I can use any one of them to estimate the rate of heat energy.

2. When I discussed things with him last, he mentioned to me that just one temperature sensor at the beginning of the exhaust pipe is good to estimate the energy of the gas. Now this equation in this page is same as equation (3) I believe :

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heatra.html#c2

If my sensor is outside the pipe, I would only obtain T_cold, how can I estimate the rate of energy change without knowing T_hot?

Chestermiller
Mentor
Chestermiller, I really really appreciate your help. The thing is I am not able to communicate effectively with the professor who is supposed to be helping me through this. At the same time, its pretty late to be changing what I am doing and he always seems busy to respond.

1. If Qa is the same in equation (2) and (3) I assume I can use any one of them to estimate the rate of heat energy.
If I recall correctly, you are trying to determine the amount of energy exiting the engine and entering the exhaust pipe.
2. When I discussed things with him last, he mentioned to me that just one temperature sensor at the beginning of the exhaust pipe is good to estimate the energy of the gas.
Your professor is correct. The rate of energy loss in the gas stream entering the exhaust pipe is wCp(Tbeginning-Tref), where w is the mass rate of flow through the pipe, Cp is the heat capacity of the gas, and Tref is the reference temperature from which enthalpy of the gas stream is calculated.

The equations you are using are trying to estimate the heat loss between the entrance and exit of the exhaust pipe. But this is not what you are interested in determining. You are interested in determining the rate at which energy enters the exhaust pipe.

Chet

you are trying to determine the amount of energy exiting the engine and entering the exhaust pipe.
Yes sir, that is right.

The rate of energy loss in the gas stream entering the exhaust pipe is wCp(Tbeginning-Tref)
Wouldn't I need another sensor to calculate w (mass flow rate)? I was hoping to use equation (3) because it uses constants belonging to the pipe which I can obtain easily I think.

Given that I have my temperature sensor outside the pipe at the beginning of the exhaust, is my temperature sensor sensing Tbeginning or Tref?

Chestermiller
Mentor
Yes sir, that is right.

Wouldn't I need another sensor to calculate w (mass flow rate)? I was hoping to use equation (3) because it uses constants belonging to the pipe which I can obtain easily I think.

Given that I have my temperature sensor outside the pipe at the beginning of the exhaust, is my temperature sensor sensing Tbeginning or Tref?
Hi Jay,

This is an engine, and you are trying to determine the energy losses from the engine. Is this correct?

If so, here are some questions:

Are you a chemical engineer, an automotive engineer, a mechanical engineer, or a physicist?
What year are you in at school?
Have you had a course in Thermo yet?

What is the rate at which gasoline is consumed?
What is the air/fuel mass flow ratio?
How does the mass flow rate of gases exiting the exhaust pipe compare with the mass flow rate of fuel and air supplied to the engine? Does this give you a hint as to how to determine w?

If you have had Thermo, have you learned about enthalpy, heat of reaction, and heat of formation?
Do you know the composition of the gases in the exhaust?
Do you know the coolant flow rate to the radiator, and the temperature change of the coolant in passing through the radiator?

Chet

1 person