Question #97581
Find T(x,t) in a laterally insulated 2 m-long rod if k=10-4 m2/s and T(x,0)=100(2x-x2), T(0,t)=0=T(2,t).
1
Expert's answer
2019-11-01T10:14:28-0400

T(0,t)=0=T(2,t)T(0,t)=0=T(2,t) ----- (i)

This is the boundary condition for laterally insulated rod having ends maintained at certain temperatures.

Governing equation for the following condition

T(x,t)/t=k2T(x,t)/x2∂T(x,t)/∂t=k ∂^2T(x,t)/∂x^2 ------------- (ii)

For separation of variable T(x,t)=f(x).g(t)T(x,t) = f(x).g(t) ---------- (iii)

Substituting equation (iii) in equation (ii);

we get g(t)/g(t)=kf(x)/f(x)=lg'(t)/g(t) = k f''(x)/f(x) =l ----- (iv)

g(t)/g(t)=lg'(t)/g(t)=l

d(g(t))/g(t)=ldtd(g(t))/g(t)=ldt

Integrating both sides, we get

ln(g(t)/c2)=ltln (g(t)/c_2) = lt

g(t)/c2=exp(lt)g(t)/c_2 = exp(lt)

g(t)=c2exp(lt)g(t) = c_2 exp(lt) ----(v)

kf(x)/f(x)=lk f''(x)/f(x) =l

2f(x)/x2=(l/k)f(x)∂^2f(x)/∂x^2=(l/k)f(x)

2f(x)/x2=m2f(x)∂^2f(x)/∂x^2 =-m^2f(x)

l/k=m2l/k= -m^2 -----(vi)

Solution to this differential equation is

f(x)=c1sin(mx)+c2cos(mx)f(x) = c_1sin(mx)+c_2 cos(mx)

T(x,t)=(c2exp(lt))(c1sin(mx)+c3cos(mx))T(x,t) =(c_2 exp(lt))(c_1sin(mx)+c_3cos(mx))


Using equation (i)

T(0,t)=(c2exp(lt))(c1sin(0)+c3cos(0))=0T(0,t) =(c_2 exp(lt))(c_1sin(0)+c_3cos(0))= 0

T(0,t)=c3c2exp(lt)=0T(0,t) =c_3c_2 exp(lt)= 0

c3=0c_3 =0

Using equation (i)

T(2,t)=(c2exp(lt))(c1sin(m.2))=0T(2,t)=(c_2 exp(lt))(c_1sin(m.2)) =0

T(2,t)=c1c2exp(lt)sin(m.2)=0T(2,t)=c_1c_2exp(lt)sin(m.2)=0

We obtain 2.m=nπ2.m= nπ

m=nπ/2m=nπ/2

Using equation (vi)

l/k=m2=n2π2/4l/k=-m^2=-n^2π^2/4

l=n2π2k/4l =-n^2π^2k/4

So the equation becomes

T(x,t)=n=1(c2exp(n2π2kt/4))(c1sin(nπx/2))T(x,t) =\sum_{n=1}^{\infty}(c_2 exp(-n^2π^2kt/4))(c_1sin(nπx/2))

T(x,t)=n=1(cn(sin(nπx/2)exp(n2π2kt/4))T(x,t) =\sum_{n=1}^{\infty}(c_n(sin(nπx/2) exp(-n^2π^2kt/4))

cn=c1.c2c_n= c_1.c_2

We have to find cn now

T(x,0)=n=1(cn(sin(nπx/2)exp(n2π2k.0/4))T(x,0) =\sum_{n=1}^{\infty}(c_n(sin(nπx/2) exp(-n^2π^2k.0/4))

n=1(cn(sin(nπx/2)=100(2xx2)\sum_{n=1}^{\infty} (c_n(sin(nπx/2) = 100(2x-x^2)

It is a sine Fourier series.we know;

cn=2×02(100(2xx2)(sin(nπx/2))dxc_n = 2×\int_{0}^{2}(100(2x-x^2)(sin(nπx/2))dx

solving the integration using by parts and then putting the limits we get,

cn=3200/(n3π3)((1)n1)c_n = -3200/(n^3π^3)((-1)^n-1)

Answer:

T(x,t)=n=1(cn(sin(nπx/2)exp(n2π2kt/4))T(x,t) = \sum_{n=1}^{\infty} (c_n(sin(nπx/2) exp(-n^2π^2kt/4)),

where cn=3200/(n3π3)((1)n1)c_n = -3200/(n^3π^3)((-1)^n-1)



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