(adsbygoogle = window.adsbygoogle || []).push({}); 1-D Conduction+Convection::Deriving a Finite Difference Scheme

***For those of you who would like to skip the derivation of the energy balance, skip ahead to theboldsubheading below where the actual Finite Difference Scheme begins***

I am trying to derive a finite difference scheme for 1-Dimensional conduction and convection. I am using a similar approach to one used in my lecture notes for conduction only. It is a control volume approach to this.

I would upload a picture, but I think the 1D case is simple enough to picture. Picture a bar with constant cross-sectional area. We will work with a cross-sectional "slice" of the bar whose length isdx

EDIT: Picture added

So as said, there is conduction occurring (i.e. a temperature gradient is present) and there is convection going on. Here are the properties of the bar [with dimensions]:

A = constant cross-sectional area [Length^{2}]

K = thermal conductivity [Power/(Length-Temp)]

c = specific heat capacity [Energy/(Mass-Temp)]

h = convective coefficient of heat transfer [Power/(Length^{2}-Temp)]

[itex]\rho[/itex] = density [Mass/Length^{3}]

The Energy balance is given by:

[tex]E_{in} + E_{generated} = \Delta U +E_{out}\qquad(1)[/tex]

There is a conduction term on the left edge of the control volume atxand a conduction term at the right edge atx + dx

That is, the energy balance becomes:

[tex]q_xA\,dt+QA\,dx\,dt=\Delta U+q_{x+dx}A\,dt + q_hP\,dx\,dt \qquad (2)[/tex]

where:

[itex]q_x[/itex] is the heat conducted (heat flux) into the control volume (CV) at the surface edge x.

[itex]q_{x+dx}[/itex] is the heat conducted (heat flux) out of the control volume at the surface edge x+dx.

Qis the internal heat source (or sink); heat generated (or removed) per unit volume.

[itex]q_h = h(T-T_{\infty})[/itex] is the convective heat transfer

Pis the perimeter around the constant cross-sectional areaA

Presumably they have neglected convective heat transfer (HT) at the end faces and are only considering HT via the body's surface area *though I am not sure why ~ Any thoughts on that?).

So here goes the derivation:

first we note that the conductive term atxis given by Fourier's law of conduction:

[tex]q_x = -K_{xx}\frac{dT}{dx} \qquad (3)[/tex]

Using a two-term Taylor series, we can expand the [itex]q_{x+dx}[/itex] term to be:

[tex]

q_{x+dx} = -[K_{xx}\frac{dT}{dx}+\frac{d}{dx}(K_{xx}\frac{dT}{dx})\,dx] \qquad (4)

[/tex]

plugging (3) and (4) into (2) yields:

[tex]-K_{xx}\frac{dT}{dx}A\,dt+QA\,dx\,dt=\Delta U-[K_{xx}\frac{dT}{dx}+\frac{d}{dx}(K_{xx}\frac{dT}{dx})\,dx]A\,dt + q_hP\,dx\,dt

\qquad (5)[/tex]

Canceling the [itex]-K_{xx}\frac{dT}{dx}A\,dt[/itex] term of both sides and then dividing through by A*dx*dt and noting that [itex]\Delta U = c\rho A\,dT\,dx[/itex] and assuming thatKis constant yields:_{xx}

[tex]K\frac{\partial{}^2 T}{\partial{x}^2} + Q = \rho c \frac{\partial{T}}{\partial{t}} +\frac{hP}{A}(T - T_{\infty}) \qquad (6)[/tex]

or for no source term we have:

[tex]\frac{\partial{}^2 T}{\partial{x}^2} =\frac{ \rho c}{K}\, \frac{\partial{T}}{\partial{t}}+\frac{hP}{AK}(T - T_{\infty}) \qquad (7)[/tex]

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Now for the finite difference scheme:

If we have a one dimensional bar we want to find the temperature at a point 'p' that is located equidistant in the x-direction to points 'p+1' and 'p-1.' We can rewrite the derivative terms via Taylor expansions; that is,

[tex]\frac{\partial{}^2 T}{\partial{x}^2} \approx \frac{T(x+dx) - 2T(x) + T(x-dx)}{dx^2}

\qquad (8)[/tex]

and

[tex]\frac{\partial{T}}{\partial{t}} \approx \frac{T(t+dt)-T(t)}{dt}\qquad (9)[/tex]

Hence, (7) becomes:

[tex]\frac{T(x+dx) - 2T(x) + T(x-dx)}{dx^2}} =\frac{ \rho c}{K}\, \frac{T(t+dt)-T(t)}{dt}+\frac{hP}{AK}(T - T_{\infty}) \qquad (10)[/tex]

EQUATION (10) is where I am a little confused. I don't think that these should be "total derivatives" but instead should be "partials" but as engineers, we like to abuse notation. So I would like to fix this (any thoughts are helpful).

Looking at the right-hand-side of (10) there are 2 "T" terms I need to work out. Is the "(T+dt)-term" at location "x" ? I belive it is. And is the convection "T-term" at time "t" ? I also believe that is true.

If so, I am going to let T(x) at time "t" be represented by [itex]T_p^t[/itex] similarly T(x,t+dt) would be represented by [itex]T_p^{t+dt}[/itex] and so on.

With this notation, we can represent equation (10) as:

[tex]T_{p+1} - 2T_p + T_{p-1} =(\frac{ dx^2\rho c}{K})\, \frac{T^{t+dt}_p-T_p^t}{dt}+(\frac{dx^2\,hP}{AK})(T_p^t - T_{\infty}) \qquad (11)[/tex]

Now we have purposefully left the time dependence off of the T-terms on the left hand side as this will determine whether the scheme is explicit or implicit --that is whether it is evaluated at 't' or 't+dt'

I am still unsure of the "T" on the right-hand-side in the brackets [itex](T_p^t - T_{\infty})[/itex].

Should I have left the time dependence ambiguous on that as well? And define it when I define the scheme type? And is it indeed at point 'p' (pretty sure it is)?

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# 1-D Conduction+Convection: eriving a Finite Difference Scheme

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