What is characteristic time in Fourier number exactly?

Question

• What is characteristic time in Fourier number?

• How can I calculate characteristic time?

Suppose I started heating water in a closed container by immersion rod and temperature increases continuously. Suppose at time $t=260 sec$ temp. at center is $40^oC$, and at $t=480 sec$, center temp is $42^oC$. How do I calculate Fourier number I need characteristic time.

• Then which time I should consider, $263$ or $480 sec$?

Answer 1

Introduction

I don't like the use of the word 'characteristic' in the definition used by Wikipedia. A characteristic time scale of a process is the time scale which defines the process, i.e. in thermo-fluid flow for Peclet number $\mathrm{Pe}=vl/\alpha\ll1$, where diffusive forces dominate, the time scale has to be a diffusive one $t_d=l^2/\alpha$. On the other hand if $\mathrm{Pe}\gg1$, where inertial forces dominate, the time scale has to be an inertial one $t_i=l/v$.

The Fourier number is therefor only applicable in the diffusive regime, i.e. for transient conduction problems where $\mathrm{Pe}\ll1$. Then it relates the process time $t_p$ to the characteristic diffusive time scale $t_d$ such that the Fourier number is the dimensionless ratio of time scales: $$\mathrm{Fo}=\frac{t_p}{t_d}$$

Often the Fourier number is used to separate two different regimes in transient conduction problems:

1. short-term (aka penetration theory) - relates to the situation where conduction has just started and heat hasn't penetrated through-out the system yet; this is characterized by the growing of a thermal boundary layer $\delta\left(t\right)=\sqrt{\pi\alpha t_p}$. We speak of penetration theory when the boundary layer has reached approximately half way ($\delta\left(t\right)<L/2$) through the system. This means that it is relevant for $\mathrm{Fo}<\frac{1}{4\pi}\approx0.1$.
2. long-term - For $\mathrm{Fo}\gg0.1$, the thermal boundary layer has grown through-out the system.

It is important to separate these regimes because solving the resulting physical problem requires different methods. For short-term conduction, the temperature profiles look similar in time except for a scaling factor (which happens to be the boundary layer), such problem can be solved using similarity solutions. For long-term transient conduction, the temporal temperature rise is uncoupled from the spatial variation and it is possible to solve using separation of variables.

Answer to the question

The process time $t_p$ which is required in the Fourier number to characterize the temperature rise from $40$ to $42$ degrees celsius is the amount of time it took to accomplish the temperature rise, i.e. $480-260=220$ seconds.

Answer 2

Fourier number is a measure of heat penetration depth. Together with Biot number it characterizes transient aspect of heat conduction. Characteristic time here is the time it takes for temperature difference between the object and the surrounding to drop to $1 \over e$ (37%) of its value. If initial temperature is $T_0$, and surrounding temperature is $T_{\infty}$ at characteristic time this holds:

${{T(\tau)-T_{\infty}}\over{T_0 - T_{\infty}}} = {1 \over e}$

For more info I suggest looking at chapter 6 of Fundamentals of Heat and Mass Transfer by Kothandaraman