8B (Differential Equations)

Put

θ = T - T_{r}

and

u = U - T_{r}

. The term

- a θ

models oven cooling to the room,

Q

is heater input, and

- b u = b θ - b u

makes the pizza relax toward the oven temperature. Both protocols supply unit total heat because

\int δ (t) d t = 1

and the rectangular pulse has height

1/ τ

and width

τ

For the impulse,

θ

jumps by one at zero while

u

remains continuous:

θ_{1} (t) = e^{- a t}, u_{1} (t) = \frac{b}{b - a} (e^{- a t} - e^{- b t})

for

a \neq = b

; when

a = b

u_{1} = b t e^{- b t}

G_{a} (t) = H (t) \frac{1 - e ^{- a t}}{a}, K (t) = H (t) \frac{b}{b - a} (\frac{1 - e ^{- a t}}{a} - \frac{1 - e ^{- b t}}{b}) .

The rectangular-pulse solutions are

θ_{2} (t) = \frac{G _{a} ( t ) - G _{a} ( t - τ )}{τ}, u_{2} (t) = \frac{K ( t ) - K ( t - τ )}{τ} .

Finally

T_{i} = T_{r} + θ_{i}

U_{i} = T_{r} + u_{i}

. These formulas make

T_{2}, U_{2}

continuous at

0, τ

; only

T_{1}

jumps at the delta impulse. As

τ \to 0

G_{a}^{'} = θ_{1}

and

K^{'} = u_{1}

, because the rectangular pulse is an approximate identity.

 Ancestors (9)

There are no discussions about this article yet.

There are currently no matching articles.