Temperature of fluid flowing through pipe

[Chestermiller]

Hey Chet,

I was studying the three constants from this video. Is the "L" of the Nusselt number the same as the diameter, it seems to be the thickness of the boundary layer. Please verify this. In another pdf, it seems to be the same as the diameter. In wikipedia under "laminar flow" it says diameter again.

But in this video he seems to say "L" is a small layer. Go to 5:40. Have we done it correctly?

Yes. We have done it correctly.

The length scale that you use in the Nusselt number and/or the Reynolds number depends on the context. Sometimes you use a variable distance z to represent the length scale if the heat transfer coefficient is varying with position, and sometimes you even can use a boundary layer thickness. In any event, since these results are all either correlations of experimental data (in the case of turbulent flow) or analytic solutions to the fluid mechanics and energy equations (in the case of laminar flow), as long as things are done in a consistent manner, there is no problem.

Chet

 

[Chestermiller]

Okay. I have to make this into a MATLAB code/model. A model is just a bunch of equations right? I am good with MATLAB, but I need the "model", if I understand it correctly its the equations.

Yes. There's the math model that consists of the set of equations you are solving, and there's the computer model, which represents the computer code you are using to solve the equations.

Okay to find the thermal conductivity at 650 C, I am using this link :

http://bouteloup.pierre.free.fr/lica/phythe/don/air/air_k_plot.pdf

650 C = 923 K, using this value in the equation pasted :

k (@ 923K) = 0.0641756431 ≈ 0.0642

So using Nu = 24, L = 0.07, k = 0.0642, I get

h = 22.01143

This comes in the range of forced convection for gases. I calculate the temperature again using this?

No yet. To use your equation, we need to determine the heat transfer coefficient h2 on the outside of the tailpipe.

See this link: http://www.engineeringtoolbox.com/convective-heat-transfer-d_430.html. You will notice that they give an equation for the heat transfer coefficient in air flow past an object (in our case, the tailpipe): h = 10.45 - v + 10 v^1/2. It would be very convenient if we could use this equation for h2, since it already has the air physical properties inherently built into the correlation. However, as you can see, there is no length scale in the correlation, so I am uncomfortable with it, and don't know how accurate it would be for your situation. If it were me working on this, I would test this equation out against the correlations in BSL for heat transfer in flow past a sphere and over a cylinder to see how it compares with these confirmed correlations. I would like to know if there is any range of diameters for either case in which this equation gives accurate predictions. However, that's just me.

If you are comfortable using this equation, then you can use it to get h2. But, don't expect it to be as accurate as the results for h1, considering the above caveats. My feeling is that the heat transfer coefficient h2 is going to depend on some length scale, and there is going to be a difference between cross flow and axial flow. In your case, the air flow is axial along the outside of the tailpipe. I also think that h2 may vary with the placement of your measuring device. For example it may vary with how far back the device is from the exit of the muffler.

Is there any way of calibrating your system, by doing some tests where you measure both the temperature on the tailpipe (your main experimental method) and also in the exhaust gas leaving the tailpipe (temporary set up)? This would help calibrate the value of h2.

If you are comfortable using the toolbox equation to get h2, then we are close to the end of our journey. You just substitute the speed of the car for v in the equation.

We are going to do this iterative process for every data point of temperature? I am going to be having like 1800 data points, given that I am sampling every second in a half an hour drive. Does doing the iterative process for each data point sound reasonable?

Are you required to provide the results on streaming data, or are you allowed to record the data during the test and analyze them afterwards? If it is the latter, then no problem. Even if it's the former, there still shouldn't be a problem, considering how fast computers are now-a-days.

Chet

 

[Jay_]

If you are comfortable using the toolbox equation to get h2, then we are close to the end of our journey. You just substitute the speed of the car for v in the equation.

I am tempted to complete this problem quickly. But I took a look at that function in a grapher. It attains a local maxima at v = 25 m/s and becomes zero before v = 120 m/s, past this it becomes negative. The graph shown in the website also seems to be for values before 20 m/s, so I guess its been modeled for that range (v < 20 m/s). Since I will be driving the car at speeds of up to 80 mph (35.76 m/s), maybe its not a good idea. If there are other models on the heat transfer coefficient of air, you could tell me about those. But its okay if its not there. Let's proceed to do it the same way. I think only the temperature would change instead of the 650 C, it would be 35 C. Right?

Are you required to provide the results on streaming data, or are you allowed to record the data during the test and analyze them afterwards? If it is the latter, then no problem.

Its the latter.

 

[Jay_]

Would you be able to find any papers that could give an equation? I search and got one on thermal conductivity. But we don't want that.

 

[Chestermiller]

I am tempted to complete this problem quickly. But I took a look at that function in a grapher. It attains a local maxima at v = 25 m/s and becomes zero before v = 120 m/s, past this it becomes negative. The graph shown in the website also seems to be for values before 20 m/s, so I guess its been modeled for that range (v < 20 m/s). Since I will be driving the car at speeds of up to 80 mph (35.76 m/s), maybe its not a good idea. If there are other models on the heat transfer coefficient of air, you could tell me about those. But its okay if its not there. Let's proceed to do it the same way. I think only the temperature would change instead of the 650 C, it would be 35 C. Right?

We can't use the same correlation for outside the tube that we did for inside the tube, because the gas flow outside is different. Let's look at 2 things:

1. Let's try completing the problem using the equation in toolbox. I just want you to get some experience. We already assumed a car speed of 35 mph for determining the gas flow inside the tube. Let's now assume a 35 mph air flow rate outside the pipe, since this is the relative velocity of the air with respect to the tube. Use the toolbox equation to get h2, and then use your equation to get the exhaust gas temperature for some assumed measured value of the wall temperature (say 350 C?). See what you come up with. Or, if the exhaust gas temperature actually was 650 C, what would that imply for the value of the measured wall temperature?

It might be possible that the ratio of h1 to h2 may not vary much with the car speed. If this were the case, then we may be able to extrapolate to higher car speeds. We can try some other car speeds to see what we get.

2. As far as the heat transfer coefficient h2 outside the tube is concerned, we an get an upper bound to this heat transfer coefficient by assuming that the air flow is perpendicular to the tailpipe, rather than parallel (as in our real world case). We can get this upper bound by using the correlation in BSL for flow over a cylinder. We can then compare this upper bound with the values that we get from the toolbox equation (for various values of the relative air velocity). Maybe we can deduce an equivalent diameter for the tube that will give the same value as for the toolbox equation. This also might enable us to extrapolate to higher car speeds.

What I'm saying here is that we have to "play with" the correlations and numbers until we arrive at something that is comfortable for us. This is called "doing scouting calculations."

Also, see Eqns. 11-13 of this reference: https://www.google.com/search?q=hea...ome..69i57.18410j0j1&sourceid=chrome&ie=UTF-8

Chet

 

[Jay_]

Hey Chet,


I am using these equations :

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

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

We calculated h1 (inside) = 22.01143

Now using the equation in engineering toolbox, and the velocity as 35 mph = 15.6464 m/s.

Using this I get h2 = 34.3591

Putting this in equation above I get :


T_{gas} = \frac{(T_{otp} - T_{amb})*22}{34}+T_{inp}

So for example if the ambient temperature is 35 C, the pipe outside is 150 C. And we assume the same temperature inside, we have

Temperature of gas = 224 C.

Sound reasonable?

 

[Chestermiller]

Hey Chet,


I am using these equations :

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

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

We calculated h1 (inside) = 22.01143

Now using the equation in engineering toolbox, and the velocity as 35 mph = 15.6464 m/s.

Using this I get h2 = 34.3591

Putting this in equation above I get :


T_{gas} = \frac{(T_{otp} - T_{amb})*22}{34}+T_{inp}

So for example if the ambient temperature is 35 C, the pipe outside is 150 C. And we assume the same temperature inside, we have

Temperature of gas = 224 C.

Sound reasonable?

No. The starting equation should be:
\frac{T_{gas} - T_{inp}}{T_{otp} - T_{amb}} = \frac{h2}{h1}
So the final equation should be:
T_{gas} = \frac{(T_{otp} - T_{amb})*34}{22}+T_{inp}

Also, why don't you try a higher temperature for the wall temperature than 150 C, like 250C? I would like to show you how the iteration works for correcting the gas temperature.

I feel like we're getting closer to where we want to be now. Are you starting to feel more comfortable? But, we still have more work to do.

Have you had a chance to try out that correlation for flow over a cylinder to assess the upper bound for h2?

Chet

 

[Jay_]

Have you had a chance to try out that correlation for flow over a cylinder to assess the upper bound for h2?

I haven't tried that yet. I will though.

Are you starting to feel more comfortable? But, we still have more work to do.

Yea I do. But to get the equation right, its :

T_{gas} = \frac{(T_{otp} - T_{amb})*34}{22}+T_{inp} and assuming outer wall temperature for 250 C, and the inner wall temperature being the same, and the ambient 35 C, we get

T_gas = 582.27 C

Now, I assume we start from post # 53

STEPS FOR GAS TEMPERATURE ITERATION

1) Find μ at given temperature (now it is 582 C) (http://www.lmnoeng.com/Flow/GasViscosity.php)
2) Use Re = 4W/(μπD)
3) Find the corresponding ordinate value in the graph of BSL from Re (M say)
4) Find the corresponding Prandtl number (http://www.mhtl.uwaterloo.ca/old/onlinetools/airprop/airprop.html)
5) Use the relation below to find the Nusselt number (I have a question on this *):

Nu=MRePr^{1/3}(\frac{μ_b}{μ_0})^{0.14}

6) Find the thermal conductivity (http://bouteloup.pierre.free.fr/lica/phythe/don/air/air_k_plot.pdf)

7) Find the heat transfer coefficient using

h = \frac{kNu}{L}

8) Find the heat transfer coefficient of standard ambient temperature air.

9) Find gas temperature using
T_{gas} = \frac{(T_{otp} - T_{amb})*h_{out}}{h_{in}}+T_{inp}


Repeat till different in the results of previous iterations are below the acceptable error.

* The equation we use in the step 5 seems to be slightly different in different sources. The difference being that some ignore the viscosity factor, and the number M and the power on the Reynolds number are different. What's the actual equation?

Another question, we are using air for finding the values of the constants in steps 1, 4 and 6. But the gas inside is a mix of N₂, H₂O and CO₂ and small fractions of CO, NO_X and HC. Is the value going to be roughly the same? Is it safe to make that assumption?

 

[Chestermiller]

I haven't tried that yet. I will though.



Yea I do. But to get the equation right, its :

T_{gas} = \frac{(T_{otp} - T_{amb})*34}{22}+T_{inp} and assuming outer wall temperature for 250 C, and the inner wall temperature being the same, and the ambient 35 C, we get

T_gas = 582.27 C

Now, I assume we start from post # 53

STEPS FOR GAS TEMPERATURE ITERATION

1) Find μ at given temperature (now it is 582 C) (http://www.lmnoeng.com/Flow/GasViscosity.php)
2) Use Re = 4W/(μπD)
3) Find the corresponding ordinate value in the graph of BSL from Re (M say)
4) Find the corresponding Prandtl number (http://www.mhtl.uwaterloo.ca/old/onlinetools/airprop/airprop.html)
5) Use the relation below to find the Nusselt number (I have a question on this *):

Nu=MRePr^{1/3}(\frac{μ_b}{μ_0})^{0.14}

6) Find the thermal conductivity (http://bouteloup.pierre.free.fr/lica/phythe/don/air/air_k_plot.pdf)

7) Find the heat transfer coefficient using

h = \frac{kNu}{L}

8) Find the heat transfer coefficient of standard ambient temperature air.

9) Find gas temperature using
T_{gas} = \frac{(T_{otp} - T_{amb})*h_{out}}{h_{in}}+T_{inp}


Repeat till different in the results of previous iterations are below the acceptable error.

Excellent, excellent, excellent. You're rapidly becoming a heat transfer man.

* The equation we use in the step 5 seems to be slightly different in different sources. The difference being that some ignore the viscosity factor, and the number M and the power on the Reynolds number are different. What's the actual equation?

This is all for flow in a pipe, correct? (Of course, other geometries have different relationships.) Give me some examples so I can comment better. Don't forget that, for flow in a pipe, this is just a correlation of experimental data. If the viscosity factor is ignored, then one is implicitly assuming that the temperature difference between the wall and the bulk flow is not large.

Another question, we are using air for finding the values of the constants in steps 1, 4 and 6. But the gas inside is a mix of N₂, H₂O and CO₂ and small fractions of CO, NO_X and HC. Is the value going to be roughly the same? Is it safe to make that assumption?

Not necessarily. We would like to check this out. The physical properties of mixtures of gases can be estimated using the techniques discussed in the first chapter of each section in BSL. (Momentum transport and Heat Transport). To do this, you need to know the mole fractions of the various gases in the exhaust. We're going to assume complete combustion, and no side reactions. So we need to do some stoichiometry. Let's start simple by assuming that the fuel is pure octane. Write the balanced chemical reaction equation for combustion of octane to yield water and CO2. If you have 1 mole of octane, how many moles of O2 are consumed, and how many moles of H20 and CO2 are produced? How many moles of N2 end up in the exhaust gas? What are the mole fractions of N2, H20, and CO2 in the exhaust gas. To check to see how good an approximation using octane is, calculate the air to fuel mass ratio and compare it with 14.7 (our assumed ratio). If it's close to this value, using the calculated mole fractions will be adequate.

Chet

 

[Jay_]

Give me some examples so I can comment better.

Check the "Dittus-Boelter" relation here in the very end of the document:

http://www.me.umn.edu/courses/old_me_course_pages/me3333/gallery/eq 3.pdf

I found another link (I can't locate it now), which had different equations for turbulent flow in a circular pipe for different range of Reynolds numbers.

The physical properties of mixtures of gases can be estimated using the techniques discussed in the first chapter of each section in BSL. (Momentum transport and Heat Transport).

Like in example 1.4-2 I guess (for viscosity). But then what does the viscosity factor become?

Also, there is example 1.4-1 in which they find the viscosity at various temperatures for CO₂. It looks like this uses other constants characteristic to the gas. How do we know these values for the gas in the exhaust?

We're going to assume complete combustion, and no side reactions. So we need to do some stoichiometry. Let's start simple by assuming that the fuel is pure octane. Write the balanced chemical reaction equation for combustion of octane to yield water and CO2. If you have 1 mole of octane, how many moles of O2 are consumed, and how many moles of H20 and CO2 are produced?

But shouldn't we have done this when we assumed the constants of the gas? We picked all the constants for air.

I have literature from Volkswagen that shows the percentages in the exhaust emissions by mass are as follows :

N₂ = 71%
CO₂ = 14%
H₂O = 13%
CO, HC, NO_X = 2% (I assume all of this to be CO)

This is by mass. Now, by my calculations on 100 g of this exhaust gas. We have the following:

Moles of each gas

moles of N₂ = 71/28 ≈ 2.5357
moles of CO₂ = 14/44 ≈ 0.3182
moles of H₂O = 13/18 ≈ 0.7222
moles of CO = 2/28 ≈ 0.0714

Total moles : 3.6475

Mole fractions

mole fraction of N₂ = 2.5357/3.6475 ≈ 0.6952
mole fraction of CO₂ = 0.3182/3.6475 ≈ 0.0872
mole fraction of H₂O = 0.7222/3.6475 ≈ 0.1980
mole fraction of CO = 0.0714/3.6475 ≈ 0.0196

 

[Chestermiller]

Check the "Dittus-Boelter" relation here in the very end of the document:

http://www.me.umn.edu/courses/old_me_course_pages/me3333/gallery/eq 3.pdf

I found another link (I can't locate it now), which had different equations for turbulent flow in a circular pipe for different range of Reynolds numbers.

Like I said, these correlations are based on experimental data. They are what the individual authors judged as the best fit to the data. They will all give very similar predictions. In the case of the viscosity factor, as I said, leaving that out is equivalent to assuming that the temperature difference between the bulk and the wall is small.

Like in example 1.4-2 I guess (for viscosity). But then what does the viscosity factor become?

The viscosity factor is the ratio of the viscosity at the tube wall temperature to the viscosity at the bulk gas temperature (to the 0.14 power).

In another chapter, they have the mixing rules for thermal conductivity. Of course, for heat capacity, the mixture rule is just mole fraction.

Also, there is example 1.4-1 in which they find the viscosity at various temperatures for CO₂. It looks like this uses other constants characteristic to the gas. How do we know these values for the gas in the exhaust?

These constants are molecular properties characteristic of the particular substance, and are independent of the temperature and pressure.

But shouldn't we have done this when we assumed the constants of the gas? We picked all the constants for air.

Yes, but I wanted you to see some results, even if they are not very accurate (yet). And, I wanted to bring you along gradually, so that all the complexity was not included at one time. I have lots of experience doing modelling, and I have found great value in starting simple and seeing results early on. I have found this approach to be very effective. My three basic principles for doing modelling are:
1. Start simple
2. Start simple
and
3. Start simple

Why? If you can't solve a simple version of your problem, then you certainly won't be able to solve it in full complexity. Plus, you get to see some results right away. Plus, you will get to see what each successive refinement makes in the final results.

I have literature from Volkswagen that shows the percentages in the exhaust emissions by mass are as follows :

N₂ = 71%
CO₂ = 14%
H₂O = 13%
CO, HC, NO_X = 2% (I assume all of this to be CO)

This is by mass. Now, by my calculations on 100 g of this exhaust gas. We have the following:

Moles of each gas

moles of N₂ = 71/28 ≈ 2.5357
moles of CO₂ = 14/44 ≈ 0.3182
moles of H₂O = 13/18 ≈ 0.7222
moles of CO = 2/28 ≈ 0.0714

Total moles : 3.6475

Mole fractions

mole fraction of N₂ = 2.5357/3.6475 ≈ 0.6952
mole fraction of CO₂ = 0.3182/3.6475 ≈ 0.0872
mole fraction of H₂O = 0.7222/3.6475 ≈ 0.1980
mole fraction of CO = 0.0714/3.6475 ≈ 0.0196

OK. This is good. Now, if you have data on the individual species (like N2 from toolbox), you don't need to use the estimation procedures in BSL for those species. Of course, this is better than using the estimation procedures. However, you will still need to apply the mixing rules.

Chet

 

[Chestermiller]

In post #70, you gave VW data on the composition of the exhaust gas. What does this data imply about the air:fuel ratio? I calculate about 9.6. What do you calculate?

Assuming complete combustion of octane, I get an air to fuel ratio of 15.1, and the following mole percentages in the exhaust:

N2 73.4
CO2 12.5
H20 14.1

Chet

 

[Jay_]

Like I said, these correlations are based on experimental data. They are what the individual authors judged as the best fit to the data. They will all give very similar predictions. In the case of the viscosity factor, as I said, leaving that out is equivalent to assuming that the temperature difference between the bulk and the wall is small.

Thats okay. Could you tell me what equation we should use here. I need to get done with this, I am sorry if I sound like I am not interested in knowing more.

The thing is this due by September. And I need to get started with the actual experiment of getting the data from a car. My professor has been asking me for the 'model' before he goes on to allow us to show him the experimental data, and I was hoping this discussion would come to an end soon. I certainly do appreciate all your help. This would have been impossible without you.

But I want to get to final the answer (like I typed in post # 70) in steps.

What does this data imply about the air:fuel ratio? I calculate about 9.6. What do you calculate?

I am not sure how to go from mole fractions to the air fuel ratio. Given that we assume a air:fuel ratio of 14.7:1 for the Toyota Corolla. I would like to work the other way, going from air:fuel ratio to mole fractions. Could you show me how you calculated the air:fuel ratio from the percentages?

I would like to show him some sort of code (MATLAB) of the model, by Wednesday - day after. We can certainly fine tune the model after that.

But I want to show him one complete iteration (with the gas constants, not the air constants) soon. I keep telling him I am almost done with it and I may have given him the impression that I am doing nothing. Could we get the constants of the gas considering 14.7 air:fuel ratio and finish the iterations?

I won't be able to type the MATLAB code until I know the exact process for each.

We need to get three constants : viscosity, Prandtl number and thermal conductivity. How do we get these knowing the gas mixture?

 

[Chestermiller]

I am not sure how to go from mole fractions to the air fuel ratio. Given that we assume a air:fuel ratio of 14.7:1 for the Toyota Corolla. I would like to work the other way, going from air:fuel ratio to mole fractions. Could you show me how you calculated the air:fuel ratio from the percentages?

Given the time constraints, let's skip the VW data and how to go from the percentages to the air:fuel ratio. Let's start out by assuming that octane is a surrogate for your fuel. The balanced chemical reaction equation for the complete combustion of octane is:

C₈H₁₈+12.5 O₂ = 8 CO₂ + 9 H₂O

The number of moles of N2 in the air that enters along with this amount of O2 is 12.5x79/21=47.0, where 79 and 21 are the mole percents of N2 and O2 in the air. So the total number of moles of air that are required for complete combustion of 1 mole of octane is 59.5. The molecular weight of air is 29, and the molecular weight of octane is 114. So the air:fuel ratio is 59.5x29/114 ~ 15.1 This is close enough to 14.7. So for every mole of fuel that enters, there are 47 moles of N2, 8 moles of CO2, and 9 moles of water as the exhaust. This gives the mole fractions I showed in my previous post.

I would like to show him some sort of code (MATLAB) of the model, by Wednesday - day after. We can certainly fine tune the model after that.

I don't know how to program in MATLAB, so I can't help you there. FORTRAN is what I always use.

But I want to show him one complete iteration (with the gas constants, not the air constants) soon. I keep telling him I am almost done with it and I may have given him the impression that I am doing nothing. Could we get the constants of the gas considering 14.7 air:fuel ratio and finish the iterations?

We need to get three constants : viscosity, Prandtl number and thermal conductivity. How do we get these knowing the gas mixture?

Regarding the Prantdl number, that's the heat capacity times the viscosity divided by the thermal conductivity. So, you really need to get those. You start out by getting the properties of the pure substances. You look up heat capacity vs temperature in a handbook for each species. You can get the viscosity and thermal conductivity of each species either from literature equations that were fit to data or using the estimation procedures in chapters 1 and 9 in BSL. Once you know the properties of the pure species, you use the mixing rules in chapters 1 and 9 to get the properties of the mixtures (except for heat capacity, which is weighted in terms of mole fraction).

Chet

 

[Jay_]

That's okay Chet. I am pretty good with MATLAB. I will show some calculations based on this post of yours soon, so that I will have something to show him. Let me then get your approval on a flowchart diagram before I get to the code. :-)

There is also one more thing. The final quantity we want to measure is Q (rate of energy) of the gas. So for that we need mass flow rate (which we calculate), Cp (specific heat capacity of gas, which we also calculate) and temperature of gas (which we again find out by iteration).

But what does ΔT represent? You mentioned ΔT = Tgas - Treference, but on what basis do we choose the reference point and if it is any random point, how is it possible that a gas has two energy values, because if we change the reference value we would get difference heat exchange values.

 

[Chestermiller]

There is also one more thing. The final quantity we want to measure is Q (rate of energy) of the gas. So for that we need mass flow rate (which we calculate), Cp (specific heat capacity of gas, which we also calculate) and temperature of gas (which we again find out by iteration).

But what does ΔT represent? You mentioned ΔT = Tgas - Treference, but on what basis do we choose the reference point and if it is any random point, how is it possible that a gas has two energy values, because if we change the reference value we would get difference heat exchange values.

I don't know what is in your professor's mind as far as his definition of what constitutes the heat contained in the exhaust gases. What might make sense is to calculate the enthalpy of the exhaust gases relative to what it would be if we cooled them down to the ambient temperature. That may be what he's thinking.

Chet

 

[Jay_]

What might make sense is to calculate the enthalpy of the exhaust gases relative to what it would be if we cooled them down to the ambient temperature. That may be what he's thinking

So that makes it ΔT = Tgas - Tambient. Okay. I will show him all this in code tomorrow. Thanks Chet. I will get back to you for help. :-)

 

[Jay_]

The three constants - viscosity, heat capacity and thermal conductivity.

Now there are two places to go: pure substance to pure substance at given temperature and from there to the mixture at given temperature.

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

Lets do heat capacity first. I can use these links for the pure substances.

http://www.engineeringtoolbox.com/nitrogen-d_977.html >> for N2
http://www.engineeringtoolbox.com/carbon-dioxide-d_974.html >> for CO2
http://www.engineeringtoolbox.com/water-vapor-d_979.html >> H2O
http://www.engineeringtoolbox.com/carbon-monoxide-d_975.html >> CO

Since, I am using a 2001 Corolla, I think there would be some CO emissions? If I assume complete combustion, I wouldn't have any CO correct? Do you think I should then go with some literature data or calculate the value myself? My real question is can we assume complete combustion with a kind of old car?

Then I use the mole fraction mixing concept.

Handling viscosity next. BSL gives me the following: (Table 1.1-3)

N2 (@ 20 C) = 0.0175 mPa.s
H2O (@ 100 C) = 0.01211 mPa.s
CO2 (@ 20 C) = 0.0146 mPa.s

I know there is a Sutherland's relation to the equation I found in wikipedia, but its range is only 0 K to 555 K, the gas temperatures are going to be beyond that. So are you aware of any relation?

Also, I can't find thermal conductivity at different temperatures, so I thought I could just use the Prandtl number. Since you mentioned the Prandtl number is largely insensitive to temperature, its value is the same at any temperature. So I could find that, and then work backwards to find the thermal conductivity using

k = Cp*mu/Pr

After I get it for the individual ones at the given temperature, I would use the mixing rules. Would that be correct?

 

[Chestermiller]

Since, I am using a 2001 Corolla, I think there would be some CO emissions? If I assume complete combustion, I wouldn't have any CO correct? Do you think I should then go with some literature data or calculate the value myself? My real question is can we assume complete combustion with a kind of old car?

Considering that you are assuming an air:fuel ratio of 14.7, and the complete combustion relation gives a value of 15.1, I wouldn't worry too much about the CO. Also, the amount is going to be small, so how much of a difference could it make. Leaving it out is just equivalent to using the remaining gases as a surrogate for CO. This is not going to be the largest inaccuracy in your calculation. Also, whatever you're doing is certainly better than using properties for air.

Then I use the mole fraction mixing concept.

Handling viscosity next. BSL gives me the following: (Table 1.1-3)

N2 (@ 20 C) = 0.0175 mPa.s
H2O (@ 100 C) = 0.01211 mPa.s
CO2 (@ 20 C) = 0.0146 mPa.s

I know there is a Sutherland's relation to the equation I found in wikipedia, but its range is only 0 K to 555 K, the gas temperatures are going to be beyond that. So are you aware of any relation?

Also, I can't find thermal conductivity at different temperatures, so I thought I could just use the Prandtl number. Since you mentioned the Prandtl number is largely insensitive to temperature, its value is the same at any temperature. So I could find that, and then work backwards to find the thermal conductivity using

k = Cp*mu/Pr

You can find all the actual data you need in Chemical Engineers' Handbook, Perry, Chapter 3 for the pure gases. It gives viscosities and thermal conductivites on the pure gases up to even higher temeratures than you need. See the section of transport properties. It gives equations for the heat capacity of each gas as a function of temperature in an earlier section. All this is in tables (or, in the case of N2 and CO2 viscosity, alignment chart).

After I get it for the individual ones at the given temperature, I would use the mixing rules. Would that be correct?

Sure.

 

[Jay_]

Chemical Engineers' Handbook, Perry, Chapter 3

Okay. I got a pdf copy of the book. Now coming to mixing of viscosity of gases, the example 1.4-2 in BSL is not very clear. How is the last column of the table computed?

In any case, after that how do I go to the value at a given temperature?

 

[Chestermiller]

Okay. I got a pdf copy of the book. Now coming to mixing of viscosity of gases, the example 1.4-2 in BSL is not very clear. How is the last column of the table computed?

You already calculated ø_αβ in the previous column, so now you do a weighted sum over the mole fractions. (You hold α constant, and sum over the β's).

In any case, after that how do I go to the value at a given temperature?

The tables in Perry give the pure gas parameter values as a function of temperature. For any temperature you want, you either interpolate in the tables or use an equation fit to the temperature dependence of the parameters.

Chet

 

[Jay_]

The tables in Perry give the pure gas parameter values as a function of temperature.

The thing that is going to change in every iteration is the temperature. Mole fractions will remain constant. So we could just come up with a formula for the exhaust right? I didn't think the method would be so complicated. Because if we are going to do this calculation in each iteration, its going to be a lot of work.

Coming to the example and the last column. From the formula mentioned above the last column, this is the number I come up with the for the first element of the last column (which they have as 0.763) :

(0.133)*(1.000 + 0.730 +0.727) + (0.039)*(1.000 + 0.730 + 0.727) + (0.828)*(1.000 + 0.730 + 0.727) = 2.457

Nor is it this way that : (0.133)*(1.000 + 0.730 + 0.727) = 0.326781

How do they come up with 0.763?

 

[Chestermiller]

The thing that is going to change in every iteration is the temperature. Mole fractions will remain constant. So we could just come up with a formula for the exhaust right? I didn't think the method would be so complicated. Because if we are going to do this calculation in each iteration, its going to be a lot of work.

I don't see what's so complicated. Also, you have been concerned about the accuracy of the results. You can do some preliminary calculations to determine how far off you would be if you neglected the temperature iteration, and just assumed some nominal temperature for all cases.

Coming to the example and the last column. From the formula mentioned above the last column, this is the number I come up with the for the first element of the last column (which they have as 0.763) :

(0.133)*(1.000 + 0.730 +0.727) + (0.039)*(1.000 + 0.730 + 0.727) + (0.828)*(1.000 + 0.730 + 0.727) = 2.457

Nor is it this way that : (0.133)*(1.000 + 0.730 + 0.727) = 0.326781

How do they come up with 0.763?

(0.133)(1)+(0.039)(0.73)+(0.828)(0.727)

Chet

 

[Jay_]

I don't see what's so complicated.

Sorry, I am just getting lazy maybe lol

I see how the numbers come up. Thanks Chet. But in that case shouldn't the formula above the column be
∑(β = 1 to 3) x_β*ø_αβ.

The subscript on x should be β (because β is changing) and not α, right?

 

[Jay_]

Okay. I also checked chapter 9, example 9.3-3 on the thermal conductivity of gas mixes. Now, the last thing is the specific heat capacity of the mixture.

How do we get to that? Is it by weighted mole fractions again, or weighted masses?

 

[Chestermiller]

Sorry, I am just getting lazy maybe lol

I see how the numbers come up. Thanks Chet. But in that case shouldn't the formula above the column be
∑(β = 1 to 3) x_β*ø_αβ.

The subscript on x should be β (because β is changing) and not α, right?

In my book, the subscript on x is β.

Chet

 

[Chestermiller]

Okay. I also checked chapter 9, example 9.3-3 on the thermal conductivity of gas mixes. Now, the last thing is the specific heat capacity of the mixture.

How do we get to that? Is it by weighted mole fractions again, or weighted masses?

The specific heat of an ideal gas mixture is a straight weighted sum over mole fraction.

Chet

 

[Jay_]

Okay. I need to cite a source for each equation. Which source does this equation come from? :

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

I don't see that exact equation in BSL.

 

[Chestermiller]

This equation is derived from q = h1 (Tgas - Twall)=h2(Twall-Tair). It says that the heat flux from the gas to the wall is equal to the heat flux from the wall to the air. All the heat flow from the gas to the air passes through the wall and the resistances are in series .

Chet

 

[Jay_]

Okay. One more thing. How do we obtain the constants for the outside of the pipe (of standard air)? You mentioned BSL section on submerged objects.

We can find the thermal conductivity using this equation :
http://bouteloup.pierre.free.fr/lica/phythe/don/air/air_k_plot.pdf

From BSL would I be using equation 14.4-7 and 14.4-8 to find the Nusselt number (since it is a cylinder), and then find the value of 'h'? In doing so what would be the value of the characteristic length L as we go to find h from Nu?