The discussion centers on the mathematical relationship for temperature change in fluids flowing through pipes, specifically in contexts such as exhaust pipes and radiator hoses. Key parameters influencing temperature change include cross-sectional diameter, viscosity, volumetric flow rate, specific heat, length of the pipe, density, and thermal conductivity. Participants recommend consulting "Transport Phenomena" by Bird, Stewart, and Lightfoot for detailed equations and methodologies. The complexity of heat transfer in these systems necessitates a thorough understanding of both fluid dynamics and thermodynamics.
PREREQUISITESEngineers, thermodynamics students, and professionals involved in automotive engineering or fluid mechanics who are looking to optimize heat transfer in piping systems.
You can always put the data in as discrete values in a table that you can interpolate.
No. It doesn't seem reasonable. What were your values of h1 and h2?Jay_ said:In the code I have for now, I am using 0.004 as the proportionality constant. However, there is another issue. The temperature of gas variable seems pretty high.
I put dummy values :
RPM : 88
temperature of pipe wall : 125 C (398 K)
ambient temperature : 35 C (308 K)
mileage : 28 mpg
x : 10 cm (0.1 m)
d : 7 cm (0.07 m)
I get the gas temperature as 1135 K (862 C). Are these values reasonable? If the temperature is 862 C, I imagine the pipe would start to melt?
Use 0.004 from 4000 to 10000, and 0.003 at 60000, with a linear fit on the log of the ordinate between 10000 and 60000.Jay_ said:But what is this linear variation exactly? What is the equation for it?
The graph in BSL doesn't have a very good resolution. The line of the sketch is not fine, its pretty thick and the ordinate axis is not calibrated.
Use 0.004 from 4000 to 10000, and 0.003 at 60000, with a linear fit on the log of the ordinate between 10000 and 60000.
Yes.Jay_ said:Okay. So that fixes it.
Yes, just to see.Regarding your second paragraph - are you saying I need to calculate the Reynolds number, Nusselt number and heat transfer coefficient for the inside the same way I did for the outside and see what results I get.
In this case, I would need the density of the gas mixture because the Reynolds equation requires it. And the velocity would be the velocity of the gas?
YesAnd then I use 0.332(Re^0.5)(Pr^(1/3)) ?
It's very hard to tell without seeing the details of the calculations. One way of checking, of course, is to do the calculation by hand, which shouldn't take too much effort. So bite the bullet.Jay_ said:I input the car speed. 88 RPM is slow, but a car can move at that speed right? In mph this speed is 6.8, which is indeed very slow. I increased the RPM to 400, which makes the speed in in mph as 30.94 (tire diameter is 26 inches).
Its hard to show the whole code. But I have attached a screen shot of the values (all are supposed to be in S.I. units). The ones that look odd, should probably tell me where I made a mistake?
I have attached a word file to this. The funny thing is, when I keep the pipe temperature the same (125 C) and increase the speed of the car (from 88 rpm to 400 rpm), the gas temperature inside decreases! I also found that this 'saturates' to the value of the pipe temperature (125 C), if I increase the speed indefinitely. I made rpm 1010 and then 1020 and found it comes to the pipe temperature in both instances.
In any case, below (in the attached .docx file) is the data I got for 400 rpm. Everything should be in S.I. units. So could you see which is odd?
rpm = 400; %input('Enter RPM data :\n');
T_pipe = 125; %input('\nEnter pipe temperature in deg. celsius :\n');
d = 7; %input('\nEnter diameter of pipe in centimeters : \n');
x = 10; %input('\nEnter distance of sensor from pipe end in centimeters : \n');
T_amb = 35; %input('\nEnter ambient temperature in deg. celsius :\n');
mpg = 28; %input('\nEnter mileage for the given drive cycle in miles per gallon: \n');
D = 26; %input('\nEnter the diameter of the tire in inches: \n');
D = D*2.54/100;
d = d/100;
x = x/100;
T_pipe = T_pipe + 273.15;
T_amb = T_amb + 273.15;
v = pi*D*rpm/60;
mph = 2.23694*v;
V_fuel = mph/(3600*mpg);
M_fuel = 6.073*V_fuel;
m_gas = 0.45359*15.7*M_fuel;
T_gas = 2.5*T_pipe;
T_gas_est = 0;
while (abs(T_gas - T_gas_est) > 0.2)
T_gas_est = T_gas;
Tf_in = mean([T_gas_est T_pipe]);
Tf_out = mean([T_amb T_pipe]);
sig_phi = SIGMA_PHI(Tf_in); % I can post HOW these functions are defined if needed
u_gas = VISCOSITY_GAS(Tf_in, sig_phi); % SIGMA_PHI and all these are based from the mixing
k_gas = THERMAL_GAS(Tf_in, sig_phi); % rules in BSL
cp_gas = SPECIFIC_HEAT_GAS(Tf_in);
Re_in = (4*m_gas)/(u_gas*pi*d);
Pr_in = cp_gas*u_gas/k_gas;
Nu_in = 0.004*(Re_in)*(Pr_in^(1/3)); % Find equation for ordinate!
h_in = Nu_in*k_gas/d;
rho_air = DENSITY_AIR(Tf_out); % I can post HOW these functions are defined if needed
u_air = VISCOSITY_AIR(Tf_out);
k_air = THERMAL_AIR(Tf_out);
cp_air = SPECIFIC_HEAT_AIR(Tf_out);
Re_out = (v*rho_air*x)/u_air;
Pr_out = (cp_air*u_air)/k_air;
Nu_out = 0.332*(Re_out^(1/2))*(Pr_out^(1/3));
h_out = Nu_out*k_air/x;
T_gas = (h_out/h_in)*(T_pipe - T_amb) + T_pipe;
endrpm = 455; %input('Enter RPM data :\n');
T_pipe = 250; %input('\nEnter pipe temperature in deg. celsius :\n');
d = 7; %input('\nEnter diameter of pipe in centimeters : \n');
x = 10; %input('\nEnter distance of sensor from pipe end in centimeters : \n');
T_amb = 35; %input('\nEnter ambient temperature in deg. celsius :\n');
mpg = 28; %input('\nEnter mileage for the given drive cycle in miles per gallon: \n');
D = 26; %input('\nEnter the diameter of the tire in inches: \n');
D = D*2.54/100;
d = d/100;
x = x/100;
T_pipe = T_pipe + 273.15;
T_amb = T_amb + 273.15;
v = pi*D*rpm/60;
mph = 2.23694*v;
V_fuel = mph/(3600*mpg);
M_fuel = 6.073*V_fuel;
m_gas = 0.45359*15.7*M_fuel;
T_gas = 2.5*T_pipe;
T_gas_est = 0;
while (abs(T_gas - T_gas_est) > 0.01)
T_gas_est = T_gas;
Tf_in = mean([T_gas_est T_pipe]);
Tf_out = mean([T_amb T_pipe]);
sig_phi = SIGMA_PHI(Tf_in);
u_gas = VISCOSITY_GAS(Tf_in, sig_phi);
k_gas = THERMAL_GAS(Tf_in, sig_phi);
cp_gas = SPECIFIC_HEAT_GAS(Tf_in);
Re_in = (4*m_gas)/(u_gas*pi*d);
Pr_in = cp_gas*u_gas/k_gas;
Nu_in = 0.004*(Re_in)*(Pr_in^(1/3)); % Find equation for ordinate!
h_in = Nu_in*k_gas/d;
rho_air = DENSITY_AIR(Tf_out);
u_air = VISCOSITY_AIR(Tf_out);
k_air = THERMAL_AIR(Tf_out);
cp_air = SPECIFIC_HEAT_AIR(Tf_out);
Re_out = (v*rho_air*x)/u_air;
Pr_out = (cp_air*u_air)/k_air;
Nu_out = 0.332*(Re_out^(1/2))*(Pr_out^(1/3));
h_out = Nu_out*k_air/x;
T_gas = (h_out/h_in)*(T_pipe - T_amb) + T_pipe;
endHi Jay,Jay_ said:Calculations :
This is the simple code I put in MATLAB :
Code:rpm = 400; %input('Enter RPM data :\n'); T_pipe = 125; %input('\nEnter pipe temperature in deg. celsius :\n'); d = 7; %input('\nEnter diameter of pipe in centimeters : \n'); x = 10; %input('\nEnter distance of sensor from pipe end in centimeters : \n'); T_amb = 35; %input('\nEnter ambient temperature in deg. celsius :\n'); mpg = 28; %input('\nEnter mileage for the given drive cycle in miles per gallon: \n'); D = 26; %input('\nEnter the diameter of the tire in inches: \n'); D = D*2.54/100; d = d/100; x = x/100; T_pipe = T_pipe + 273.15; T_amb = T_amb + 273.15; v = pi*D*rpm/60; mph = 2.23694*v; V_fuel = mph/(3600*mpg); M_fuel = 6.073*V_fuel; m_gas = 0.45359*15.7*M_fuel; T_gas = 2.5*T_pipe; T_gas_est = 0; while (abs(T_gas - T_gas_est) > 0.2) T_gas_est = T_gas; Tf_in = mean([T_gas_est T_pipe]); Tf_out = mean([T_amb T_pipe]); sig_phi = SIGMA_PHI(Tf_in); % I can post HOW these functions are defined if needed u_gas = VISCOSITY_GAS(Tf_in, sig_phi); % SIGMA_PHI and all these are based from the mixing k_gas = THERMAL_GAS(Tf_in, sig_phi); % rules in BSL cp_gas = SPECIFIC_HEAT_GAS(Tf_in); Re_in = (4*m_gas)/(u_gas*pi*d); Pr_in = cp_gas*u_gas/k_gas; Nu_in = 0.004*(Re_in)*(Pr_in^(1/3)); % Find equation for ordinate! h_in = Nu_in*k_gas/d; rho_air = DENSITY_AIR(Tf_out); % I can post HOW these functions are defined if needed u_air = VISCOSITY_AIR(Tf_out); k_air = THERMAL_AIR(Tf_out); cp_air = SPECIFIC_HEAT_AIR(Tf_out); Re_out = (v*rho_air*x)/u_air; Pr_out = (cp_air*u_air)/k_air; Nu_out = 0.332*(Re_out^(1/2))*(Pr_out^(1/3)); h_out = Nu_out*k_air/x; T_gas = (h_out/h_in)*(T_pipe - T_amb) + T_pipe; end
Its pretty straight forward what we did, but merely put as code. Now the functions for obtaining the physical parameters of the gas and that of air, I took from the sources we discussed and I have them as polynomials in temperature. That's why I put the input as Tf_in (for the gases) and Tf_out (for air), these are basically the film temperatures.
Also, I changed the temperature from 125 C to 250 C, like we did earlier and I got a temperature of 527 C for the gas. Seems okay to me. Do you feel its all right?
You don't need my help to answer this. Just do the calculations by hand. It should only take about 20 minutes, and then you'll be sure.Jay_ said:Okay. But do you think the values are not correct? I would only need to troubleshoot if that's the case right?
Well done!Jay_ said:The values seem correct to me. I did the calculations by hand before and verified them in the code (like the mixing rules for instance). But since I haven't done this before, I wanted you to check if the values are inside the typical values.
Jay_ said:Hey Chet,
Congratulations on becoming Mentor, you have certainly been mine through this, and I will mention you the report.
I am back here because I am still feeling dicey about the equation for the Nusselt number inside the pipe.
In the pdf attached, they seem to use the Seider-Tate relation (Page 3, eq. (1)). I would have overlooked it, but its the proportionality constant values. We are using 0.003 to 0.004, and they Seider-Tate uses a value as high as 0.027, almost 6 to 9 times as high!
Since, we have put all this effort, I don't want us to mess up on this one aspect. You mentioned that these equations are based off on collected data, but I can't imagine why the proportionality constants would vary so much.
The difference in Nu is only 7%. This compares with the 20-30% uncertainty cited in BSL for these correlations.Jay_ said:Wikipedia says it must be above 10,000. Let's consider that Re = 10000.
Taking the log of the Nusselt number equations in both cases :
BSL : log(Nu) - (1/3)*log(Pr) = log(0.004) + log(10,000) = 1.60206
Seider-Tate : log(Nu) - (1/3)*log(Pr) = log(0.027) + 0.8*log(10,000) = 1.63136
Is that difference acceptable?
The disturbing thing about that graph in BSL is :
1. The line of the graph is thick, leading to inaccuracies
2. At Re = 60,000 the line is actually below 0.003, which is another inaccuracy. At Re = 10,000 also, its below 0.004 - that means the Nusselt numbers are smaller than calculated - that would have been comforting. But the problem being that the Nu by Seider-Tate comes to be bigger. If it came to be smaller, I could have thought of it as the inaccuracy in the line of the graph (and my vision).
Yes. But, check BSL's description to be sure.-------------------------------------------------------------------------------------
Also!
Since I have temperature based polynomials for the viscosity of the gas, I could as well include the viscosity factor in the calculations right?
viscosity factor = (mu_bulk/mu_boundary_layer)
Correct?
In my opinion, again, you're splitting hairs. The correlation in BSL appears in many other books, and BSL indicate that it is based on Seider and Tate's data. They even give an equation with a 0.027 instead of 0.026. Any of these choices that you describe is adequate, and they all have about the same degree of accuracy.Here, the temperature of the bulk gas is the T_gas_estimated, and the denominator is the INNER film temperature right? I am just going to include the (viscosity factor)^0.14 in the Nusselt number equation.
And which would be a better pick, Seider-Tate or BSL? If Sieder-Tate is more accurate, I want to use BSL for Nusselt numbers up to the value of 10,000 and then use Sedier-Tate from there on. But which is more accurate in your opinion?