Question

Consider a long, thin, uniform rod of constant cross-section whose temperature distribution is θ(x,t), surrounded by an atmosphere of constant uniform temperature θ0. The sides of the rod are not insulated so that heat is lost from the longitudinal surface at a rate h(θ(x,t)- θ0) per unit length, for some constant h. Derive the equation which describes the temperature distribution in the rod and show that it can be written

∂φ/∂t=k/ρc (∂^2 φ)/(∂x^2 ) - (h/ρAc) φ

where φ(x,t) = (θ(x,t)- θ0), k is the thermal conductivity, ρ the density , c the thermal capacity and A the cross-sectional a

Answer 1

Consider a long, thin, uniform rod of constant cross-section whose temperature distribution is θ(x,t), surrounded by an atmosphere of constant uniform temperature θ0. The sides of the rod are not insulated so that heat is lost from the longitudinal surface at a rate h(θ(x,t)- θ0) per unit length, for some constant h. Derive the equation which describes the temperature distribution in the rod and show that it can be written

∂φ/∂t=k/ρc (∂^2 φ)/(∂x^2 ) - (h/ρAc) φ

where φ(x,t) = (θ(x,t)- θ0), k is the thermal conductivity, ρ the density , c the thermal capacity and A the cross-sectional area of the rod. (You may assume one-dimensional heat flow in the rod.)

Such a rod of length l is initially maintained at a constant temperature θ1. Both ends of the rod are then brought to and maintained at temperature θ0 and the rod is surrounded by an atmosphere at temperature θ0. Show that the temperature distribution in the rod at any subsequent time t has the form
θ(x,t)= θ_0+∑_(n=1)^∞▒〖B_n e^(-k/ρc ((n^2 π^2)/l^2 +h/KA)t ) Sin (nπx/l)〗

and determine Bn. (When attempting to solve the equation above you may find it easier to swap the φ term over to the other side of the equation.)

Answer 2

Yeah i think i made a typo. so change it to -

Answer 3

ok

Answer 4

do you how do second part of qs?

Answer 5

Know*

Answer 6

Did you try working it out first?>

Answer 7

ya i try alot, do not know how and where to start.

Answer 8

please give me any clue? im really fed up with this qs.

Answer 9

On a stretch ofsingle-lane road with no entrances or exits the traffic density r(x,t)is a continuous function of distance xand time t, for all t > 0, and the traffic velocity u(r) is a function of density alone.
Two alternative models are proposed to represent u:

(i) u= uSL (1 – r^n/ r^nmax)
where n is a positive constant
(ii) u= 1/3 uSL In (rmax/r)where uSLis the maximum speed limit on the road and rmax is themaximum density of traffic possible on the road (corresponding tobumper-to-bumper traffic).

a) Evaluate the maximum rate of traffic flow for cases (i)and (ii) above. Show that for both cases the maximum rate of traffic flow isless than uSLrmax and that for case (i)it increases towards this upper limit as nbecomes very large.
b) It is assumed that a model of the form given in case (i)is a reasonable representation of actual traffic behaviour. Apply the method ofcharacteristics to analyse the following situation. A queue of cars is stoppedat a red traffic light on a road for which the maximum speed limit is 40 m.p.h.It may be assumed that the queue is very long and that the road ahead of thelight is empty of traffic. The light turns green and remains green for only 45seconds. If a car which is initially a quarter of a mile behind the light getsthrough before the light changes back to red determine the smallest integervalue that n can have. (Hint: showinitially that the car will not start moving until a certain time after thelight has turned green, and then solve the appropriate differential equationfor the position of the car.)