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Index/Differential equations/ODE-21

ODE-21

Newton cooling

Lumped thermal ODE.

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GoverningNewton cooling

Governing equation

T=Ta+(T0−Ta)e−ktT=T_a+(T_0-T_a)e^{-kt}

t∈[0.2,60]t \in \left[0.2, 60\right]

where

Ta
Ta (°C)
T0
T0 (°C)
k
k (1/min)
t
t (min)
T
Newton cooling (°C)

The figure, part by part

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ODE-21 · phase
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①

Governing law — red

T=Ta+(T0−Ta)e−ktT=T_a+(T_0-T_a)e^{-kt}

t∈[0.2,60]t \in \left[0.2, 60\right]

t=0.2t = 0.2T=79.05T = 79.05

②

Domain of the variable — blue

tmin⁡=0.2,tmax⁡=60t_{\min} = 0.2,\quad t_{\max} = 60

t∈[0.2,60]t \in \left[0.2, 60\right]

t=0.2t = 0.2t=0.2t = 0.2

③

Response curve — green

T=Ta+(T0−Ta)e−ktT=T_a+(T_0-T_a)e^{-kt}

t∈[0.2,60]t \in \left[0.2, 60\right]

t=0.2t = 0.2T=79.05T = 79.05

④

Rate of change — amber

dTdt\frac{\mathrm{d}T}{\mathrm{d}t}

t∈[0.2,60]t \in \left[0.2, 60\right]

t=0.576t = 0.576T′=−4.584T^{\prime} = -4.584

Lecture brief

Historical brief

Exponential decay, the harmonic oscillator, SIR epidemics, Lotka–Volterra and Bessel are the first solvable ODEs of nature and populations. The lab is rate, period and phase portrait in closed form. This sheet (ODE-21 — Newton cooling) is the form associated with Newton cooling. Working symbols: TaTa, T0T0, kk, tt →\rightarrow TT. Lumped thermal ODE. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute TT from TaTa, T0T0, kk, tt in Differential equations via T=Ta+(T0−Ta)e−ktT=T_a+(T_0-T_a)e^{-kt} Lumped thermal ODE. Use it when a real differential equations question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given Ta=20.000 ∘CTa = 20.000\,\mathrm{^{\circ}C}, T0=80.000 ∘CT0 = 80.000\,\mathrm{^{\circ}C}, k=0.080 1/mink = 0.080\,\mathrm{1/min}, t=10.000 mint = 10.000\,\mathrm{min}, the governing relation T=Ta+(T0−Ta)e−ktT=T_a+(T_0-T_a)e^{-kt} yields T=46.960 ∘CT = 46.960\,\mathrm{^{\circ}C}. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Newton cooling T46.960 °C
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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Lumped thermal ODE. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube