I am trying to solve a heat transfer problem related to calculating the cooling rate after quenching a sample.
The equations (and explanations) are taken from this paper: https://sci-hub.se/10.1029/jb091ib04p0d509. The equations to solve are:
where Rn
is defined as:
The major problem I am having right now is that I have no idea how to implement in Mathematica an "nth root".
Question:
- How to implement in Mathematica an "nth root" to calculate
Rn
?. - How can I calculate the cooling rate equation (equation 8) to get any of the cooling rates from Table 2 below:
Here's my (no working) code with the parameters to obtain the first cooling rate value of the table (260):
density = 2; (*g/cm3*)
Cp = 0.25; (*cal/g*C*)
k = 0.05; (*cal/cm*s*C*)
h = 0.072;(*Heat transfer coefficient for waterin cal/cm2*s*C*)
L = 0.144; (*cm*)
x = 0;
mass = 25(*mg*)
Tsample = 620;(*C*)
Tmedium = 23; (*C*)
alfa = k/(density*Cp);
Fo = alfa*t/L^2
Bi = h*L/k;
Rn = Bi*Cot[Rn]
n = 10000
Do[
rate = (-4 \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(i =
1\), \(k\)]\(\((Rn*Sin[Rn]*Cos[Rn*x])\)/\((Rn +
Sin[Rn]*Cos[Rn])\)\)\))*Exp[-Rn*Fo]; (*equation 7*)
, {k, 1, n} ]
crate(*equation 8*)= (Tsample - Tmedium )(*equation 9*)*
rate(*equation 7*)*alfa/L^2(*equation 10*)