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[SOLVED] Why is alpha mentioned?

dwsmith

Well-known member
Feb 1, 2012
1,673
Why is $\alpha$ mentioned? I don't see an $\alpha$.


With proper non-dimensionalization and assuming for convenience that the ratio of convection and conduction parameters $h/k = 1$, the cooling process is described by the following equations:
\begin{alignat*}{4}
T_t & = & T_{xx} & \\
T_x(0,t) & = & 0 & \\
T_x + T & = & 0 & \quad \text{at} \ x = 1\\
T(x,0) & = & f(x) &
\end{alignat*}
Here, $x$ and $t$ are dimensionless positions and times wherein the original interval has been mapped from $[0,L]$ to $[0,1]$ and a time-scale based on the diffusivity $\alpha$ has been used; also, the temperature definition is such that the ambient temperature has a reference value of zero.
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Why is $\alpha$ mentioned? I don't see an $\alpha$.


With proper non-dimensionalization and assuming for convenience that the ratio of convection and conduction parameters $h/k = 1$, the cooling process is described by the following equations:
\begin{alignat*}{4}
T_t & = & T_{xx} & \\
T_x(0,t) & = & 0 & \\
T_x + T & = & 0 & \quad \text{at} \ x = 1\\
T(x,0) & = & f(x) &
\end{alignat*}
Here, $x$ and $t$ are dimensionless positions and times wherein the original interval has been mapped from $[0,L]$ to $[0,1]$ and a time-scale based on the diffusivity $\alpha$ has been used; also, the temperature definition is such that the ambient temperature has a reference value of zero.
Hi dwsmith, :)

Well, the conduction timescale is based on diffusivity \((\alpha)\). That is,

\[t_{0}^{c}=\frac{L^2}{\alpha}\]

where \(t_{0}^{c}\) is the initial time. This is what is meant by "......time-scale based on the diffusivity $\alpha$ has been used......"

Reference: Analytical Heat Transfer by Mihir Sen (Page 6)

Kind Regards,
Sudharaka.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Hi dwsmith, :)

Well, the conduction timescale is based on diffusivity \((\alpha)\). That is,

\[t_{0}^{c}=\frac{L^2}{\alpha}\]

where \(t_{0}^{c}\) is the initial time. This is what is meant by "......time-scale based on the diffusivity $\alpha$ has been used......"

Reference: Analytical Heat Transfer by Mihir Sen (Page 6)

Kind Regards,
Sudharaka.
So I can just solve this problem without having to worry about it then, correct?
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
So I can just solve this problem without having to worry about it then, correct?
Yes, solving the partial differential equation doesn't involve \(\alpha\) since it's related to the scale used for measuring time.