# Heat and Temperature Equations-Separation of variables

#### grandy

##### Member

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#### Sudharaka

##### Well-known member
MHB Math Helper
Hi grandy,

Welcome to MHB! I think you have been given how to start this problem in the heading; Separation of Variables. So initially what you have to do is write the function $$\psi$$ as,

$\psi(x,t)=A(x)B(t)$

A good introduction about the method of separation of variables can be found >>here<<.

Kind Regards,
Sudharaka.

#### grandy

##### Member
Consider an element of the rod having length dx and situated at x:
1. Rate of heat loss by this element to the surroundings = h.φ.dx ; so that
Heat loss to surroundings in time dt = h.φ.dx.dt
Change in temperature due to heat loss to surroundings in dt = h.φ.dx.dt/(ρ.A.c.dx)

2. Rate of heat loss by conductivity along the rod = [dφ(x+dx)/dx – dφ(x)/dx].k.A
Change in temperature of the element in time dt = [dφ(x+dx)/dx – dφ(x)/dx].dt.k.A/(ρ.A.c.dx)]

3. Total change in temperature dφ = [dφ(x+dx)/dx – dφ(x)/dx].dt.k.A/(ρ.A.c.dx)]-h.φ.dt/(ρ.A… giving:
∂φ/∂t = [dφ(x+dx)/dx – dφ(x)/dx].k/(ρ.c.dx)]-h.φ/(ρ.A.c)
Now [dφ(x+dx)/dx – dφ(x)/dx]/dx = ∂²φ/∂x² giving finally:
∂φ/∂t = (k/ρc).∂²φ/∂x² - φ.h./(ρ.A.c)

I did the first part a) would please check my answer and confirm me the result, Now would you please help me with the second part. I looked at the separation of variables but I was unable to do this one because it is tough for me. Your help is really appreciated.

#### grandy

##### Member
I try to show that θ(x,t)= θ_0+∑_(n=1)^∞▒〖B_n e^(-k/ρc ((n^2 π^2)/l^2 +h/KA)t ) Sin (nπx/l)〗by looking at paul online notes but I was unsucessful. And also i was unable to find the value of Bn too. This qs is impossible tough., Please help me.

#### grandy

##### Member
now i have finished the first and second of this qs. for the third part of qs, I have got the new boundary condition but again im not sure . would please help me.

In that case the BC at L/2 would become
∂θ(L/2,t)/∂x = 0, meaning it is like an insulated end with no heat flow across it.

After that I dnt know how to do?