Help solving a fourth order polynominal

MathN00b

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Joined
Apr 16, 2018
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3
Hello all,

I am trying to use equation 7 & 8 from a fire safety engineering journal to determine the transmittance of a radiant heat flux through the atmosphere - https://ws680.nist.gov/publication/get_pdf.cfm?pub_id=909662

The math seems quite simple and is a fourth order polynomial equation. I have created a spreadsheet to solve the equations.

However, when I have entered the example given in the journal of using an air temperature of T 288K and a relative humidity of 51%, flame temperature of 1300K and path of 200m into my spreadsheet I cannot end up with a transmittance of0.75 as given in the example.

Could someone please advise what they end up with for a0, a1, a2, a3, a4 so ifI can confirm my results. Otherwise, I am happy to provide a link to my spreadsheet if you would like to review.

Any help is much appreciated.

Regards.
 

stapel

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I am trying to use equation 7 & 8 from a fire safety engineering journal to determine the transmittance of a radiant heat flux through the atmosphere - https://ws680.nist.gov/publication/get_pdf.cfm?pub_id=909662

The math seems quite simple and is a fourth order polynomial equation. I have created a spreadsheet to solve the equations.

However, when I have entered the example given in the journal of using an air temperature of T 288K and a relative humidity of 51%, flame temperature of 1300K and path of 200m into my spreadsheet I cannot end up with a transmittance of0.75 as given in the example.

Could someone please advise what they end up with for a0, a1, a2, a3, a4 so ifI can confirm my results. Otherwise, I am happy to provide a link to my spreadsheet if you would like to review.
Please reply by posting the equation here, and showing your work that led to your answer. Thank you! ;)
 

MathN00b

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Joined
Apr 16, 2018
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3
Please reply by posting the equation here, and showing your work that led to your answer. Thank you! ;)
Sorry for the delayed response.

The constants for the equation are in the following table

table.PNG

The equation for a1, a2, a3, a4 is as follows

Capture1.PNG

Contestants from the table above
Ta = 15
Ts = 1023
o = 51

Then the fourth order polynomial is
Capture2.PNG

where L = 200

I have calculated everything in an excel spreadsheet and will provide a link to it to show my working tomorrow.
 

j-astron

Junior Member
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Jan 10, 2018
Messages
181
Hi MathN00b

You're not "solving" the fourth-order polynomial. That implies finding out where it is equal to 0, which is a very hard problem. You're simply evaluating it at a specific L.

Here is some Python code I wrote up to compute the transmittance from the information in the paper:

Code:
import numpy as np # numerical Python routines                             
                                                                           
T_a = 288.0 # air temp in kelvins                                          
T_s = 1300.0 # flame temp in kelvins                                       
                                                                           
L = 200.0 # distance source to detector in m                               
phi = 0.51 # relative humidity                                             
                                                                           
Ncoeffs = 5 # number of a_n values                                         
                                                                           
# Convert the coefficients in Table 1 of Fire Safety Journal 37 (2002)     
# 181-190 into a 5 x 4 array:                                              
                                                                           
C = np.array([ [1.486, -2.003e-3, 4.68e-5, -6.052e-2],                     
               [1.225e-2, -5.900e-5, 1.66e-6, -1.759e-3],                  
               [-1.489e-4,  6.893e-7, -1.922e-8, 2.092e-5],                
               [8.381e-7, -3.823e-9, 1.0511e-10, -1.166e-7],               
               [-1.685e-9, 7.637e-12, -2.085e-13, 2.350e-10]               
             ])                                                            
                                                                           
# Empty array of coefficient values:                                       
a = np.zeros(Ncoeffs)                                                      
                                                                           
# Compute the polynomial coefficients (Eq. 8 of paper):                    
# Note that Python uses zero-indexed, row-major ordering                   
for n in range(Ncoeffs):                                                   
                                                                           
    a[n] = C[n][0] + C[n][1]*T_a + C[n][2]*T_s + C[n][3]*phi               
                                                                           
    print('Coefficient a_{} = {:.3g}'.format(n, a[n]))                     
                                                                           
# Compute the transmittance (Eq. 7 of the paper):                          
tau = a[0] + a[1]*L + a[2]*L**2 + a[3]*L**3 + a[4]*L**4                    
                                                                           
print('The calculated transmittance is tau = {:.4g}'.format(tau))
and here is the output of that code:

Code:
Coefficient a_0 = 0.939
Coefficient a_1 = -0.00348
Coefficient a_2 = 3.53e-05
Coefficient a_3 = -1.86e-07
Coefficient a_4 = 3.63e-10
The calculated transmittance is tau = 0.7502
 

MathN00b

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Joined
Apr 16, 2018
Messages
3
Thank you, this has helped me work out where I was making a mistake. Much appreciated.
 
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