Collision Integrasls Calcution:

In this notebook, we would like to report a direct comparison with two different numerical methods of collision integrals calculation with experimental results:

Here Starts the Python Notebook:



In [1]:

    
import numpy as np
import matplotlib.pyplot as plt

First numerical method is from: (Poling et al. (2000)



In [2]:

    
### Defining constants and arrays needed for calculation
AA = 1.06036
B = 0.15610
C = 0.19300
D = 0.47635
EE = 1.03587
F = 1.52996
G = 1.76474
H = 3.89411
T_ij=np.linspace(.1,200,(200/0.05)-1)



In [41]:

    
O11_2=np.empty_like(T_ij)
O11_1 = (AA/T_ij**B)+(C/exp(D*T_ij))+(EE/exp(F*T_ij))+(G/exp(H*T_ij))



In [42]:

    
plot(T_ij, O11_1)
plt.figsize(15,10)

second numerical method is from: LBNL Paper LBNL-3567E



In [6]:

    
## Noble Gages: T_ij < 1.0
a1_noble=np.array([0.18,0,-1.20407,-9.86374,16.6295,-6.73805])
a2_noble=np.array([0,0,-0.195866,20.2221,-31.3613,12.6611])
b1_noble=np.array([0,0,10.0161,-40.0394,44.3202,-15.2912])
b2_noble=np.array([0,0,-10.5395,46.0048,-53.0817,18.8125])



In [7]:

    
## Noble Gages and poliatomic Gases: 1.0<= T_ij <= 10.0
a_lt10=np.array([0.46641,-0.56991,0.19591,-0.03879,0.00259])
b1_lt10=np.array([0.357588,-0.472513,0.0700902,0.016574,-0.00592022])
b2_lt10=np.array([0.295402,-0.510069,0.189395,-0.045427,0.0037928])



In [8]:

    
## Noble Gages and poliatomic Gases: T_ij > 10.0
a1_gt10=np.array([-33.0838,101.571,-87.7036])
a2_gt10=np.array([20.0862,56.4472,46.3130])
a3_gt10=np.array([72.1052,286.393,277.146])
a4_gt10=np.array([8.27648,17.7610,19.0573])
b1_gt10=np.array([-267.00,26700,-8.9E5])
b2_gt10=np.array([201.570,-19.2265,6.31013])
b3_gt10=np.array([174.672,-27.6938,10.2266])
b4_gt10=np.array([7.36916,-3.2955,2.33033])
c=np.array([1.0,1.0e3,1.0e5])

Code:

We will Isolate the part of the algorith reserved for Noble Gases since not relevent here



In [9]:

    
def computeQ(T_jk):
    R     = 3.3145e7
    U_0   = 0.10
    roh   = 1.0e-6
    C6    = 1.0
    a_sum = 0.0
    sum_  = 0.0
    U     = U_0*exp(-roh*R)
    alpha_= log(U_0/T_jk)
    alpha10=log(U_0/10)
    if (T_jk<=1.0):
        sum_=0.0
        for i in range(len(a1_noble)):
            a_sum=a1_noble[i]+a2_noble[i]*C6**(-1/3)
            sum_= sum_+ a_sum*(T_jk)**(i/3)
        Q_lr=1.1943*(1+sum_)*(C6/T_jk)**(1/3)
#         end if
#         if (NOBLE)
#          do i=1,6
#              b_sum=b1_noble(i)+b2_noble(i)*C6**(-1/3)
#             sum_= sum_+ b_sum*(T_jk)**(i/3)
#          end do
#          Q_lr=1.1874*(1+sum_)*(C6/T_jk)**(1/3)
#         end if
    elif (1.0<T_jk<=10.0):
        sum_=0.0
        for i in range(len(b2_lt10)):
            sum_= sum_ + b2_lt10[i]*(log(T_jk))**(i)
        Q_lr=exp(sum_)
    elif (T_jk>10.0):
        sum_=0.0
        for i in range(len(b1_gt10)):
            b_sum=b1_gt10[i]+((-1)**i)*((roh*alpha10)**(-2))*c_[i]*(b2_gt10[i]+(b3_gt10[i]/alpha10)+(b4_gt10[i]/alpha10)**2)
            sum_= sum_+ b_sum*(log(T_jk))**(-2*i)
        Q_lr=((roh*alpha_)**(2))*(0.89+sum_)
    return Q_lr



In [37]:

    
O11_2=np.empty_like(T_ij)



In [38]:

    
for j in range(len(T_ij)):
    Q=T_ij[j]
    O11_2[j]=computeQ(Q)



In [39]:

    
plot(T_ij, O11_2)
plt.figsize(15,10)

Experimental results are from: JOURNAL O F RESEA RCH of the Not iona l Bureau of Stan dard s-A. Physics and Chemistry Vol. 72A, No.4, Ju ly- August 1968

Uploading the experiemental results tables:



In [19]:

    
file00="/media/idjibril/Stock/Colusion_Integrals.cvs"



In [20]:

    
TT=np.loadtxt(file00)



In [21]:

    
## Reduced Temperature Array TT
T=TT[:,0]

## Mass Diffusivity Collision Integral array O11_exp
O11_exp=TT[:,1]

We are Now going to recompute the Collision integral obtained from the different Numerical methods (1 and 2) using The values of reduced temperature from the experiemental data:



In [25]:

    
## First Numerical Method
O11_1st = np.empty_like(T)
O11_2nd = np.empty_like(T)
O11_1st = (AA/T**B)+(C/exp(D*T))+(EE/exp(F*T))+(G/exp(H*T))
## Second Numerical Method
for j in range(len(T)):
    Q=T[j]
    O11_2nd[j]=computeQ(Q)

Let's Now Plot all and compare



In [35]:

    
plt.plot(T, O11_exp)
plt.plot(T, O11_1st)
plt.plot(T, O11_2nd)
plt.figsize(15,10)

Observation:

1- We see a good match between the experiment results and the !st numerical Method results

2- We see a good match between the experiment results and the 2nd numerical Method results only on the interval of reduced temperature T=[1,10]

Analysis

We would like to report that the experimental results with which the 1st numerical method matchs are for a specific gamma(compressibility of the mixture), consequently diversion will occure when the compressiblilty changes

On the other hand, we would like to point out that some parameters(rho and U_0) were gussed in the formulation of the 2nd numerical method, for we did not have hands on the source (O'Hara & Smith (1966)) were their values were given for financial reasons.

A better argreement between the 2nd numerical method and experiment data for all gammas maybe obtained if the correct values of rho and U_0 are chosen. We will report results on the next update.

Conclusion

This notebook is not a public claim or anything like that, we would just like to point out how disorganised and difficult access to collision integral moduling data is.

We wanted to give any interested person a feel of what to expect when dealing with collision integrals. We would like to also give a motivatioin to people with relevant background to dig into this mater and maybe provide clarity on the subject.

We are just mechanical engineering students who got interested in this subject because at some point we had to medel transport properties for chemical species involved i combustion reactions.

Collision Integrasls Calcution:

In this notebook, we would like to report a direct comparison with two different numerical methods of collision integrals calculation with experimental results:

Here Starts the Python Notebook:

First numerical method is from: (Poling et al. (2000)

second numerical method is from: LBNL Paper LBNL-3567E

Code:

We will Isolate the part of the algorith reserved for Noble Gases since not relevent here

Experimental results are from: JOURNAL O F RESEA RCH of the Not iona l Bureau of Stan dard s-A. Physics and Chemistry Vol. 72A, No.4, Ju ly- August 1968

Uploading the experiemental results tables:

We are Now going to recompute the Collision integral obtained from the different Numerical methods (1 and 2) using The values of reduced temperature from the experiemental data:

Let's Now Plot all and compare

Observation:

1- We see a good match between the experiment results and the !st numerical Method results

2- We see a good match between the experiment results and the 2nd numerical Method results only on the interval of reduced temperature T=[1,10]

Analysis

We would like to report that the experimental results with which the 1st numerical method matchs are for a specific gamma(compressibility of the mixture), consequently diversion will occure when the compressiblilty changes

On the other hand, we would like to point out that some parameters(rho and U_0) were gussed in the formulation of the 2nd numerical method, for we did not have hands on the source (O'Hara & Smith (1966)) were their values were given for financial reasons.

A better argreement between the 2nd numerical method and experiment data for all gammas maybe obtained if the correct values of rho and U_0 are chosen. We will report results on the next update.

Conclusion

This notebook is not a public claim or anything like that, we would just like to point out how disorganised and difficult access to collision integral moduling data is.

We wanted to give any interested person a feel of what to expect when dealing with collision integrals. We would like to also give a motivatioin to people with relevant background to dig into this mater and maybe provide clarity on the subject.

We are just mechanical engineering students who got interested in this subject because at some point we had to medel transport properties for chemical species involved i combustion reactions.

We hope we did not disapoint you with by the content of the current notebook, and if so, we hope to be up to your expectations in the coming update.

Best regards,

MegaCFD