9. The temperature T-T(t) of a chemical sample in a furnace at time is governed
ID: 1884745 • Letter: 9
Question
9. The temperature T-T(t) of a chemical sample in a furnace at time is governed by the initial value problem ge A/T k(T-T), T (0) = To. dt where To is the initial temperature of the sample, Tf is the temperature in the furnace, and q, k, and A are positive constants, The first term on the right side is the heat generation term, and the second is the heat loss term given by Newton's law of cooling a) What are the dimensions of the constants q, k, A? b) Reduce the problem to dimensionless form using T as the temperature scale and choosing a time scale to be one appropriate tothe case when the heat loss term is large compared to the heat generated by the reactionExplanation / Answer
9. given T = T(t)
T is temperature of furmnance in K
time t is in s
tyhen given equation is
dT/dt = qe^(-A/T) - k(T - Tf), T(0) = To
a. form the equaiton
dimensions of q = [T-1 K] [ K stands for temeprature]
dimensions of A = [K]
dimensions of k = [T-1]
b . let T/Tf = T.
dT/dt = Tf*dT./dt
-A/T = -A/T.Tf
hence
Tf*dT./dt = qe^(-A/T.*Tf) - k(T. - 1)Tf, T.(0) = To/Tf
also
dt = t*dt.
hence
dT./t*dt. = qe^(-A/T.*Tf)/Tf - k(T. - 1), T.(0) = To/Tf
qt/Tf = q.
A/Tf = A.
kt = k.
dT./dt. = q.exp(-A./T.) - k.(T. - 1)
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