Prompt: Discuss time constants

Discussion: Time constants, $\tau$, are used to describe the response of linear ordinary differential equations (ODE’s), where the time rate of change of dependent variable is a function of independent variable. In layman’s terms, the “snappiness” of the response. In proper form, the linear ODE below models many equations:

τdVdt+V=f(t).\tau \frac{d V}{d t}+V=f(t).

The time constant is whatever constant value puts the equation into this form. This is unsatisfying. However, there is a good analogue with resistor/capacitor (RC) circuits, where the time constant is the product of the two values. In electronics, the resistance is a material property but here, the resistance dependent on the system geometry. For convective heat flow, $R{conv}=\frac{1}{\overline{h} A{s}}$: $\bar{h}$, the heat transfer coefficient; $A_s$, the surface area through which heat may flow. The heat capacity $C$, is simply the total thermal energy the system contains which is $c_p$, specific heat capacity, and $M$, the total mass.

RC=McpChAsR=τlumpedR C=\underbrace{\frac{\overbrace{Mc_p}^{C}}{\overline{h} A_{s}}}_{R} =\tau_{\text {lumped}}

Response to : Joe Bostick I think

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