Temperature of fluid flowing through pipe

[Jay_]

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.

 

[Chestermiller]

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.

 

[Sunfire]

This site has temperature analysis of pipe flow:

http://sites.google.com/site/vortextubeeffect/vortex-tube-rectilinear-motion/

the pipe could be stationary or moving.

 

[Chestermiller]

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

 

[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.

 

[Jay_]

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.

 

[Sunfire]

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]

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

 

[Jay_]

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]

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

 

[Jay_]

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]

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).

 

[Jay_]

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

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 learned in school. But would it be correct to use this equation for the radiator hose and exhaust, which have fluids in motion?

Thanks.

 

[Jay_]

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

 

[Jay_]

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]

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

 

[Jay_]

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 that's 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.

 

[Jay_]

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?

 

[Chestermiller]

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?

In my judgement, these equations are not adequate for what you are trying to do. You should first take a step backwards, and just try to model the radiator as a heat exchanger. You can measure the water flow through the radiator, and you can measure the water temperatures in and out. You can also estimate or measure the rate of air flow across the radiator tubes, and the temperature of the air before it hits the radiator. Then you can use the information in BSL to estimate the heat transfer coefficient for the radiator, and determine whether this is consistent with the observed change in the water temperature. Once your model of the radiator is predictive, you can work your way back into the engine, to consider the heat transfer from the engine metal to the water circulating through the engine block. To do the engine, you would have to include the heat given up in combustion of the fuel/air mixture, and the work done by the expanding gas on the pistons. Maybe you can find a book on automotive engineering to help you analyze these things.

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

 

[Jay_]

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

 

[Chestermiller]

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

 

[Jay_]

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 anyone 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]

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 anyone 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 wC_p(T_beginning-Tref), where w is the mass rate of flow through the pipe, C_p 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

 

[Jay_]

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]

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

 

[Jay_]

I am electrical and this is far from what I have dealt with. I have mostly done programming and basic electrical circuits but never dealt with a car before. The car being used is a Honda Accord I think.

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

Just through the exhaust would be good for now.

The thing is I am capturing everything in real time - the speed using the hall sensor, the temperature of the outside of the pipe using the temperature sensor. These are the only two things I have as of now. And they get saved into text files (from the COM port of my laptop) as they are being captured. Do I need to do the same for any other parameter like mass flow rate? My professor asks me to find the equation but I can't find any equation which allows me to estimate the energy coming from the exhaust by merely knowing the temperature of the outside of the exhaust pipe.

My question is do I need another sensor (to measure another varying parameter) to estimate the energy coming into the exhaust?

I have studied enthalpy, heat of reaction and heat of formation at a theoretical level. But the issue is capturing all this in real time and having the data values logged. I don't know the composition of the gases in the exhaust, but I don't think finding that would be too hard.

Thanks for your help Chet.

 

[Chestermiller]

I am electrical and this is far from what I have dealt with. I have mostly done programming and basic electrical circuits but never dealt with a car before. The car being used is a Honda Accord I think.

Just through the exhaust would be good for now.

The thing is I am capturing everything in real time - the speed using the hall sensor, the temperature of the outside of the pipe using the temperature sensor. These are the only two things I have as of now. And they get saved into text files (from the COM port of my laptop) as they are being captured. Do I need to do the same for any other parameter like mass flow rate? My professor asks me to find the equation but I can't find any equation which allows me to estimate the energy coming from the exhaust by merely knowing the temperature of the outside of the exhaust pipe.

My question is do I need another sensor (to measure another varying parameter) to estimate the energy coming into the exhaust?

I have studied enthalpy, heat of reaction and heat of formation at a theoretical level. But the issue is capturing all this in real time and having the data values logged. I don't know the composition of the gases in the exhaust, but I don't think finding that would be too hard.

Thanks for your help Chet.

I'm a little confused as to what you are trying to determine. There are two possibilities:
1. The amount of heat lost through the wall of the exhaust pipe into surrounding air
2. The amount of heat contained in the gas entering the exhaust manifold.

Heat contained in the entering gas is a relative quantity, and depends on what datum is used for the initial state. Heat lost through the wall is equal to the change in enthalpy between the inlet and exit of the pipe.

To get the heat contained in the entering gas to the manifold, you need to know the flow rate of the gas, its temperature, and its composition. You also need to know the datum for the calculation, such as the enthalpy of the air and the fuel fed to the engine.

To get the heat lost through the wall of the exhaust pipe, you need to know the average gas temperature and the heat transfer coefficient (Eqn. 3), or you need to know the inlet temperature, the outlet temperature, the gas flow rate, and the gas composition (which determines its heat capacity) (Eqn. 2). You can also get what you want if you know the inlet temperature, the heat transfer coefficient, the gas flow rate, and the gas composition (Combination of Eqns. 2 and 3). So, it isn't clear what you need to measure, and what you can calculate. The hard part is estimating the heat transfer coefficient. The easiest thing is to measure the temperatures at the inlet and the outlet, and use the (known) gas flow rate and estimated heat capacity to get the heat loss.

So, it isn't clear what your professor wants or expects. Do he even know?

Chet

 

[Jay_]

1. The amount of heat lost through the wall of the exhaust pipe into surrounding air
2. The amount of heat contained in the gas entering the exhaust manifold.

Its 2. that I actually want.

If I understand correctly, its impossible to find 2. with just a temperature sensor placed outside the exhaust pipe. Its only possible to find the heat of equation 3. with TWO temperature sensors too. Would these statements be accurate?

I am presently on search of a holy-grail-like equation which is supposed to give me 2 (heat contained in the gas) with just a temperature measurement made of the outside surface of the exhaust pipe.

The easiest thing is to measure the temperatures at the inlet and the outlet, and use the (known) gas flow rate and estimated heat capacity to get the heat loss.

What sensor could I use for that? Would you have any idea sir? I was thinking of using a exhaust gas temperature gauge but I need to find one that can be interfaced with my Arduino board.

 

[Chestermiller]

Its 2. that I actually want.

If I understand correctly, its impossible to find 2. with just a temperature sensor placed outside the exhaust pipe. Its only possible to find the heat of equation 3. with TWO temperature sensors too. Would these statements be accurate?

Heat contained in a stream is a relative quantity, so you need to define "relative to what." You might define it relative to the unburned gasoline and air fed to the engine. If you only want the heat per unit mass, then you need to know the temperature of the gas entering the exhaust manifold and its composition. If you want to know the rate of heat exiting, you also need to know its mass flow rate.

I am presently on search of a holy-grail-like equation which is supposed to give me 2 (heat contained in the gas) with just a temperature measurement made of the outside surface of the exhaust pipe.

It's not going to happen.

What sensor could I use for that? Would you have any idea sir? I was thinking of using a exhaust gas temperature gauge but I need to find one that can be interfaced with my Arduino board.

This is not my area of expertise.

Chet

 

[Jay_]

Thanks Chet. That clears a lot. I will speak to him soon and if I have any questions in regard to this I will post here again.

You have been of great help. I really appreciate it.

 

[Jay_]

Hello again Chet,

I spoke to my professor today. He said an approximate model would be fine, and I plan to use equation (2) which requires
1. mass flow rate of the gas
2. temperature of the gas
3. specific heat capacity of the gas

1. For the mass flow rate, he said it can be approximated from the speed of the vehicle because the more we hit the accelerator the more fuel is being burnt, the more gas is being thrown out of the exhaust. But is there an equation you can guide me to?

2. For the temperature of the gas, he said even though we are measuring the temperature outside the pipe. A proportionality factor can be used to estimate the temperature of the gas inside. For instance if this proportionality value is say 4, and we get a temperature reading as 150 deg C. It means the gas inside has a temperature of 600 C. I didn't ask him how we could find out the value of this proportionality, but assuming we know it. Would the outside of the pipe and the inside of the gas temperature be proportional?

3. We can find specific heat capacity from the composition the gas and the standard values on a table.

What can you say about 1. and 2. though?

 

[Chestermiller]

Hello again Chet,

I spoke to my professor today. He said an approximate model would be fine, and I plan to use equation (2) which requires
1. mass flow rate of the gas
2. temperature of the gas
3. specific heat capacity of the gas

1. For the mass flow rate, he said it can be approximated from the speed of the vehicle because the more we hit the accelerator the more fuel is being burnt, the more gas is being thrown out of the exhaust. But is there an equation you can guide me to?

2. For the temperature of the gas, he said even though we are measuring the temperature outside the pipe. A proportionality factor can be used to estimate the temperature of the gas inside. For instance if this proportionality value is say 4, and we get a temperature reading as 150 deg C. It means the gas inside has a temperature of 600 C. I didn't ask him how we could find out the value of this proportionality, but assuming we know it. Would the outside of the pipe and the inside of the gas temperature be proportional?

3. We can find specific heat capacity from the composition the gas and the standard values on a table.

What can you say about 1. and 2. though?

With regard to 1., the mass flow rate out the exhaust has to be equal to the rate of gasoline consumed plus the rate of air entering the intake manifold. If you know the mpg as a function of the vehicle speed, then you can get the gasoline consumption rate. Then, all you need to know is the air to fuel ratio.

With regard to 2., I'm not sure what he's driving at. Certainly, the temperature of the ambient air would have to be part of the picture. Maybe, the difference between the inside temperature and the air temperature will be proportional to the difference between the outside temperature and the air temperature. But, to get the proportionality constant, you would have to be able to estimate the heat transfer coefficients inside and outside the exhaust pipe.

Wish I could be more helpful.

Chet

 

[Jay_]

Chet you have been helpful, I really appreciate every post of yours.

If you know the mpg as a function of the vehicle speed, then you can get the gasoline consumption rate. Then, all you need to know is the air to fuel ratio.

Would you have an idea about such an equation?

In regards to 2. We know the temperature of the ambient air.

Maybe, the difference between the inside temperature and the air temperature will be proportional to the difference between the outside temperature and the air temperature. But, to get the proportionality constant, you would have to be able to estimate the heat transfer coefficients inside and outside the exhaust pipe.

I think that is what he meant. When you say heat transfer coefficients inside and outside the exhaust are you implying that the heat transfer coefficient of thee pipe is different for the inside of the pipe and for its outside? I assume that since its made out of the same material, it would be the same throughout.

 

[pbuk]

1. Why not start assuming a stoichiometric mix and see if the answers make sense?

2. Why not start by assuming the inside and outside temperatures are equal and see if the answers make sense? (Note that the exhaust gases have just expanded by about 10x so are not going to be anywhere near the combustion temperature).

Discuss the assumptions you have made and the results with your prof. as you go along, don't expect him to do it all for you.

 

[Chestermiller]

Would you have an idea about such an equation?

gallons per minute = (miles per minute)/(miles per gallon)

I think that is what he meant. When you say heat transfer coefficients inside and outside the exhaust are you implying that the heat transfer coefficient of thee pipe is different for the inside of the pipe and for its outside? I assume that since its made out of the same material, it would be the same throughout.

The average exhaust gas temperature (over the cross section) is higher than the temperature on the inside surface of the exhaust pipe, and the temperature on the outside surface of the exhaust pipe is higher than that of the ambient air. Each of these temperature differences is the result of a resistance to heat transfer. The first resistance is described by the heat transfer coefficient on the inside of the pipe. The second resistance is described by the heat transfer coefficient on the outside of the pipe. The ratio of the two temperature differences is equal to the ratio of the heat transfer coefficients.

Chet

 

[Jay_]

Hi Chet,

Okay. So, we have 4 temperatures :

1. temperature of gas(T_gas), 2. temperature of inside of pipe(T_inp), 3. temperature of outside of pipe(T_otp), 4. ambient temperature(T_amb)

h1 (heat transfer coefficient of inside of pipe), h2 (heat transfer coefficient of outside of pipe).

Now you have mentioned:

\frac{T_{gas} - T_{inp}}{T_{otp} - T_{amb}} = \frac{h1}{h2}

That gives me,

T_{gas} - T_{inp} = \frac{(T_{otp} - T_{amb})*h1}{h2}

I am guessing there is a relation between T_gas and T_inp based on the specific heat of the inside of the pipe or something. If that's true, then we could have an equation for just T_gas on one side and all known quantities on the other.

 

[Chestermiller]

Hi Chet,

Okay. So, we have 4 temperatures :

1. temperature of gas(T_gas), 2. temperature of inside of pipe(T_inp), 3. temperature of outside of pipe(T_otp), 4. ambient temperature(T_amb)

h1 (heat transfer coefficient of inside of pipe), h2 (heat transfer coefficient of outside of pipe).

Now you have mentioned:

\frac{T_{gas} - T_{inp}}{T_{otp} - T_{amb}} = \frac{h1}{h2}

That gives me,

T_{gas} - T_{inp} = \frac{(T_{otp} - T_{amb})*h1}{h2}

I am guessing there is a relation between T_gas and T_inp based on the specific heat of the inside of the pipe or something. If that's true, then we could have an equation for just T_gas on one side and all known quantities on the other.

No. The way to do this is to look up in the literature how to estimate the heat transfer coefficient from the bulk gas to the inside pipe wall, and the heat transfer coefficient from the outside wall to the ambient air. (It might be a reasonable approximation to neglect the heat transfer resistance of the wall, and to assume that the inside wall temperature and the outside wall temperature are equal).

Chet

 

[Jay_]

Would you be able to guide me on what sort of papers (like key words) I should search on? Also, how do you think I can get the heat transfer coefficients of the inside and outside of the pipe, would it also be available from the literature?

Also is there a source/reference for the equation I typed in post #36 of this thread. Because I need to cite that in the report I am typing.

Thank you for helping me Chet.

 

[Chestermiller]

Would you be able to guide me on what sort of papers (like key words) I should search on? Also, how do you think I can get the heat transfer coefficients of the inside and outside of the pipe, would it also be available from the literature?

Also is there a source/reference for the equation I typed in post #36 of this thread. Because I need to cite that in the report I am typing.

Thank you for helping me Chet.

What you are looking for should all be in Chapter 14 of Bird, Stewart, and Lightfoot, but I'm sure you don't want me to spoon feed it to you. You need to digest the material in the chapter. Figure 14.3-2 should have what you need to get the heat transfer coefficient inside the tailpipe. I leave it up to you to figure out what info to use to get the heat transfer coefficient outside the tailpipe.

Chet

 

[Jay_]

Hi Chet,

From what I understand from the book, the two heat transfer coefficients they call h_loc and h_in and they say (page 424) that most experimentalists use h_in which is easier to measure.

Which one is for the outside of the pipe and which is for the inside of the pipe? The comment below the graph (Fig. 14.3-2) gives h_in = (3/2)*h_loc which is useful.

I guess using this the other equation you mentioned, and which I put in post #36 can be simplified? Would that be correct?

 

[Jay_]

Hi Chet,

To clear things out on what is going on in my mind, let me put it in equations:

Okay since,

h_{in} = \frac{3}{2}*h_{loc} , and if it is like h_in is the constant for the inside of the pipe and h_loc is for the outside, we have :


\frac{T_{gas} - T_{inp}}{T_{otp} - T_{amb}} = \frac{3}{2}

That gives me :

T_{gas} - T_{inp} = \frac{3(T_{otp} - T_{amb})}{2}

and hence :
T_{gas} = \frac{3(T_{otp} - T_{amb})}{2} \ +\ T_{inp}

Here T_otp is the temperature of the outside of the pipe at the point where the exhaust starts from the engine - this is measured by my temperature sensor.

T_inp is the temperature of the inside of the pipe at the same point, which I hope to get from literature data based on the T_otp.

T_amb is the ambient temperature of the air that's easily available.

T_gas is the temperature of the gas which we can find by plugging in all the values.

Would all this be correct?

 

[Chestermiller]

Hi Chet,

To clear things out on what is going on in my mind, let me put it in equations:

Okay since,

h_{in} = \frac{3}{2}*h_{loc} , and if it is like h_in is the constant for the inside of the pipe and h_loc is for the outside, we have :


\frac{T_{gas} - T_{inp}}{T_{otp} - T_{amb}} = \frac{3}{2}

That gives me :

T_{gas} - T_{inp} = \frac{3(T_{otp} - T_{amb})}{2}

and hence :
T_{gas} = \frac{3(T_{otp} - T_{amb})}{2} \ +\ T_{inp}

Here T_otp is the temperature of the outside of the pipe at the point where the exhaust starts from the engine - this is measured by my temperature sensor.

T_inp is the temperature of the inside of the pipe at the same point, which I hope to get from literature data based on the T_otp.

T_amb is the ambient temperature of the air that's easily available.

T_gas is the temperature of the gas which we can find by plugging in all the values.

Would all this be correct?

Actually, none of it is correct. Every last bit of it is wrong. Please see my private message to you.

Chet

 

[Jay_]

Hi Chet,

I just read the chapter. Well, h_ln is the heat transfer coefficient based on the log mean temperature, and he mentions its used in most calculations. Since the h values depend on the geometry as well as the fluid properties, I would have to get that from Toyota's literature, if they have it that is.

But the Fig. 14.3-2 in Bird's book is still hard to understand. You mentioned that the gas flow in my case is turbulent, so Re > 8000. Would that be correct? So in that case I need to be looking at the right side of the graph.

On that side the Re seems to be increase for decreasing value heat transfer coefficient, because the line has a negative slope (and is not perfectly straight though). Does this go to give me a relation between the heat transfer coefficient and the flow rate, because he mentions flow rate is the Reynold's number.

I don't see the concept (equation) you mentioned in post #35, even though the term (T_b2 - T_b1)/(T₀ - T_b)_ln looks similar, but I realize it only mentions the temperature of the inside wall, and the bulk gas temperatures.

Thanks.

 

[Chestermiller]

Hi Chet,

I just read the chapter. Well, h_ln is the heat transfer coefficient based on the log mean temperature, and he mentions its used in most calculations. Since the h values depend on the geometry as well as the fluid properties, I would have to get that from Toyota's literature, if they have it that is.

It's good that your area of expertise is circuits and electronics, since that can help you greatly with your understanding of heat transfer. I assume that you are familiar with the concept of resistors in series. This is what you are dealing with in your problem. There is a resistance to heat flow from the bulk gas flow to the tube wall, and there is a second resistance to heat flow from the tube wall to the ambient air.

In the analogy between heat transfer and electronics, the temperature is analogous to voltage and the heat flux (W/m^2) is analogous to current density. The heat transfer coefficient gives the ratio of the heat flux to the temperature difference. So, it's like the reciprocal of resistance.

Do you remember studying heat conduction in freshman physics. For heat conduction through a wall, the heat flux is equal to the temperature difference times the thermal conductivity divided by the thickness of the wall. In convective heat transfer, the thickness of the wall is replaced by the thickness of the "thermal boundary layer." The heat transfer coefficient is equal to the thermal conductivity of the gas divided by the thermal boundary layer thickness (which is typically very thin).

You need to determine the two series heat transfer resistances (inside and outside the pipe wall) so that you can determine the temperature at the wall, knowing the temperature of the exhaust gas and the air temperature. This is where the correlations in BSL chapter 14 come in.

But the Fig. 14.3-2 in Bird's book is still hard to understand. You mentioned that the gas flow in my case is turbulent, so Re > 8000. Would that be correct?

Yes.

So in that case I need to be looking at the right side of the graph.

Yes.

On that side the Re seems to be increase for decreasing value heat transfer coefficient, because the line has a negative slope (and is not perfectly straight though). Does this go to give me a relation between the heat transfer coefficient and the flow rate, because he mentions flow rate is the Reynold's number.

You have to look at the total relationship. Actually, the heat transfer coefficient increases with the Reynolds number (as the flow rate increases, the thermal boundary layer gets thinner). Look at the full definition of the ordinate on the graph.

To use this correlation, you are going to have to estimate the physical properties of the gas: thermal conductivity, heat capacity, viscosity, density. You know the composition of the exhaust gas from the chemical reaction mass balance on the combustion. BSL gives relationships for estimating the thermal properties of a gas or gas mixture. You are going to be dealing primarily with a mixture of carbon dioxide, water vapor, and nitrogen. On the air side of the tailpipe, you are going to be dealing with air (a mixture of oxygen and nitrogen).

Start working on trying to estimate the heat transfer coefficient on the gas side of the wall.

Chet

 

[Jay_]

You are going to be dealing primarily with a mixture of carbon dioxide, water
vapor, and nitrogen. On the air side of the tailpipe, you are going to be dealing with
air (a mixture of oxygen and nitrogen).

Right. But I am still confused on how to establish the heat transfer coefficient on both walls of the pipe because I don't know what the ordinate of the graph represents.

Look at the full definition of the ordinate on the graph. To use this correlation, you are going to have to estimate the physical properties of the gas: thermal conductivity, heat capacity, viscosity, density. You know the composition of the exhaust gas from the chemical reaction mass balance on the combustion. BSL gives relationships for estimating the thermal properties of a gas or gas mixture.

Okay so what I am thinking :

1. Get the reaction that happens when the exhaust gas is formed
2. Calculate the mass composition of each constituent
3. Use their properties to find that of the mixture

But what does the ordinate represent exactly? From the first equation on the ordinate, I think "D" (diamter) is related to the pipe, and k, Re, Pr, mu_b, mu_0, h_ln are related to the gas. After calculating this what do I have? What does it represent?

----

Now quoting post #35 :

gallons per minute = (miles per minute)/(miles per gallon)

Here wouldn't we have to account for the air mixing with the fuel?

The data they have on the car's miles per gallon takes into consideration only the burning gallons of the fuel right?

But what comes out the tailpipe in gallons per minute is also the air-reacted carbon dioxide.

 

[Chestermiller]

Right. But I am still confused on how to establish the heat transfer coefficient on both walls of the pipe because I don't know what the ordinate of the graph represents.

Maybe it would be easier for you to work with Eqn. 14.3-16 which is equivalent to the behavior of the graph at high Re. The Nussult number is the dimensionless heat transfer coefficient. The Reynolds number is given by:

Re=\frac{ρvD}{μ}
For flow inside the tailpipe, this can be combined with the equation for the mass flow, which is given by
W=ρv\frac{πD^2}{4}
What do you get when you combine these two equations to eliminate ρv?

Okay so what I am thinking :

1. Get the reaction that happens when the exhaust gas is formed
2. Calculate the mass composition of each constituent
3. Use their properties to find that of the mixture

Basically, yes. All you need to know is the air/fuel mass flow ratio. From the internet, I found that this is about 14.7

But what does the ordinate represent exactly? From the first equation on the ordinate, I think "D" (diamter) is related to the pipe, and k, Re, Pr, mu_b, mu_0, h_ln are related to the gas. After calculating this what do I have? What does it represent?

See my comment above about using Eqn. 14.3-16. To get your feet wet, try doing the calculation assuming the gas has a viscosity equal to that of air. What would this give you for the heat transfer coefficient?

----

Now quoting post #35 :


Here wouldn't we have to account for the air mixing with the fuel?

The data they have on the car's miles per gallon takes into consideration only the burning gallons of the fuel right?

But what comes out the tailpipe in gallons per minute is also the air-reacted carbon dioxide.

Once you know the air/fuel ratio (~14.7), you have everything you need to estimate the composition of the exit gas (using the reaction stoichiometry).

To get the heat transfer coefficient for the boundary layer outside the pipe, you use the information in chapter 14 for Submerged Objects. The exhaust pipe is submerged in the air flow moving past the exhaust pipe as the car is moving. The relative velocity of the air flow is the car speed, and it is flowing parallel to the axis of the exhaust pipe (mainly).

Chet

 

[Jay_]

Hey thanks Chet. The equation I would get for the flow is :

W=\frac{ReμπD}{4}

--------------

Or are we looking for the Nusselt number?

Nu_{ln}=0.026(\frac{4W}{μπD})^{0.8}Pr^{1/3}(\frac{μ_b}{μ_0})^{0.14}

That's what I get for equation 14.3-16, eliminating ρv.

 

[Chestermiller]

Hey thanks Chet. The equation I would get for the flow is :

W=\frac{ReμπD}{4}

--------------

Or are we looking for the Nusselt number?

Nu_{ln}=0.026(\frac{4W}{μπD})^{0.8}Pr^{1/3}(\frac{μ_b}{μ_0})^{0.14}

That's what I get for equation 14.3-16, eliminating ρv.

Good. Yes, we're focusing on getting the Nussult number. This will lead directly to the heat transfer coefficient.

The first thing I would like you to do is to calculate a typical Reynolds number for the exhaust gas flow through the tail pipe. To do this, you need to get an estimate of the mass flow rate through the tailpipe. Temporarily, we can use the viscosity of air or nitrogen at the approximate temperature of the exhaust gas. We can refine the calculation later. We just want to get our feet wet and find out what "ballpark" we are playing in.

Do you have any idea how to estimate the mass flow rate of exhaust gas? If so, let's see how you do it.

Chet

 

[Jay_]

The mass flow rate of the exhaust is roughly the same as the intake of the engine. This can be calculated from the AFR (air-fuel ratio) and the concept:

gallons per minute = (miles per minute)/(miles per gallon)

Here miles per minute is measured (vehicle speed), and miles per gallon (mileage) of the car is something we will know (hopefully), so we can end up with gallons of fuel burned per minute. Using the AFR we find how much air is also used, and the exhaust flow rate has to be the combination of the two I guess. Correct?

 

[Chestermiller]

The mass flow rate of the exhaust is roughly the same as the intake of the engine. This can be calculated from the AFR (air-fuel ratio) and the concept:

gallons per minute = (miles per minute)/(miles per gallon)

Here miles per minute is measured (vehicle speed), and miles per gallon (mileage) of the car is something we will know (hopefully), so we can end up with gallons of fuel burned per minute. Using the AFR we find how much air is also used, and the exhaust flow rate has to be the combination of the two I guess. Correct?

So, let's see some numbers.

Chet