Paper 2, Section II, C

Differential Equations | Part IA, 2019

Two cups of tea at temperatures T1(t)T_{1}(t) and T2(t)T_{2}(t) cool in a room at ambient constant temperature TT_{\infty}. Initially T1(0)=T2(0)=T0>TT_{1}(0)=T_{2}(0)=T_{0}>T_{\infty}.

Cup 1 has cool milk added instantaneously at t=1t=1 and then hot water added at a constant rate after t=2t=2 which is modelled as follows

dT1dt=a(T1T)δ(t1)+H(t2)\frac{d T_{1}}{d t}=-a\left(T_{1}-T_{\infty}\right)-\delta(t-1)+H(t-2)

whereas cup 2 is left undisturbed and evolves as follows

dT2dt=a(T2T)\frac{d T_{2}}{d t}=-a\left(T_{2}-T_{\infty}\right)

where δ(t)\delta(t) and H(t)H(t) are the Dirac delta and Heaviside functions respectively, and aa is a positive constant.

(a) Derive expressions for T1(t)T_{1}(t) when 0<t10<t \leqslant 1 and for T2(t)T_{2}(t) when t>0t>0.

(b) Show for 1<t<21<t<2 that

T1(t)=T+(T0Tea)eatT_{1}(t)=T_{\infty}+\left(T_{0}-T_{\infty}-e^{a}\right) e^{-a t}

(c) Derive an expression for T1(t)T_{1}(t) for t>2t>2.

(d) At what time tt^{*} is T1=T2T_{1}=T_{2} ?

(e) Find how tt^{*} behaves for a0a \rightarrow 0 and explain your result.

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