Currently I am trying to do some experiments to try to determine thermal conductivity of my fluid which is ethanol.
To do so, I need to use the principle of TPS method which correspond to the kind of sensor I have.
I would like to plot on python ∆D(τ) function of τ and also, ∆T function of D(τ)
Basically, I have these formula which correspond to ∆D(τ).
other variable other variables
where Io is a modified Bessel function.
The paper that I am reading contains the following information that might help.
"From Eq X (D thau), we can see that the average temperature increase in the hot disk sensor is proportional to a function D(τ), which is a rather complicated function of a dimen- sionless parameter τ = √κt/a, but, numerically, it can be accurately evaluated to five or six significant figures. When using the hot disk technique to determine thermal transport properties, a constant electric current is supplied to the sensor at time t = 0, then the temperature change of the sensor is recorded as a function of time. The average temperature increase across the hot disk sensor area can be measured by monitoring the total resistance of the hot disk sensor: R = R0[1 + α ̄deltaT (t)], (28) where R is the total electrical resistance at time t, R0 is the initial resistance at t = 0, α is the temperature coefficient of resistivity, which is well known for nickel. Eq. (28) allows us to accurately determine ∆T as a function of time. If one knows the relationship between t and τ, one can plot ̄∆T as a function of D(τ), and a straight line should be obtained. The slope of that line is P0/(π3/2aK), from which thermal conductivity K can be calculated. However, the proper value of τ is generally unknown, since τ = √κt/a and the thermal diffusivity κ is unknown. To calculate the thermal conductivity correctly, one normally makes a series of computational plots of ∆T versus D(τ) for a range of κ values. The correct value of κ will yield a straight line for the ∆T versus D(τ) plot. This optimization process can be done by the software until an optimized value of κ is found. In practice, we can measure the density and the specific heat of the material separately, so that between K and κ, there is only one independent parameter. Therefore, both thermal conduc- tivity and thermal diffusivity of the sample can be obtained from above procedure based on the transient measurement using a hot disk sensor"
So if I understood I need to plot ∆T versus D(τ) from which for a certain value of characteristic time which would give me a straight line. However when I am trying to do so I will always obtain a straight line. the part that I'm not sure if the value of the modified bessel function. Please find attached my script .
from numpy import *
import scipy.special
from scipy.integrate import quad
from matplotlib.pyplot import *
def integer1(sigma):
return 1/(sigma**2)
tini = 0.015
tfin = 15
time = linspace(tini,tfin,num=1000)
n=7 # number of concentric circles of sensor
L = 1
L0 = np.i0(l*k)/(2*thau**2*n**2)
P0 = 0.1 #power
k = 1 #thermal diffusivity
a = 0.000958 # radius of biggest ring
λ = 0.169 #thermal conductivity of ethanol (im not sure if this is ok)
x=linspace(0.00000001,0.3,1000)
for K in range (0,len(x)):
# print (x[K])
theta = a**2/x[K]
Tlist = []
Dlist = []
for t in time:
thau = sqrt(t/theta)
som = 0
for l in range(L,n):
for k in range(1,n):
som += l*k*exp((-l**2+-k**2)/(4*thau**2*n**2))*L0
I = quad(integer1, 0, thau)
D = ((n*(n+1))**-2)*I[0]*som
T = (P0/(pi**(3/2)*a*λ))*D
Tlist.append(T)
Dlist.append(D)
figure(1)
plot(Dlist,Tlist)
show()
I am trying to the calculation from time 0,015 seconds until 15 seconds with 1000 points in total..0,015, 0,030, 0,045 and so on... and I for my K I am going from values of 0.00000001 until 0.3 with 1000 points in total
The paper that I am looking at is called: "Rapid thermal conductivity measurement with a hot disk sensor. Part 1. Theoretical considerations"
I hope you could help with this one.
Thank you