Please, help me with very hard problem about pipes with hot water?

Question

Two very long parallel cylindrical channles are drilled through a very large block of material. The block is so large, that it can be considered infinitely large.

http://i24.tinypic.com/2mhu73s.jpg

The diameter of the channels is 16cm, and distance between theier centers is 20cm. One cylinder contains cold +10C water, and the other contains hot +50C water. Thermal conductivity of the material is 100 mW/cm/C.

How much heat flows from hot to cold water per 1cm length?

Answer 1

The temperature T profile is governed by the Laplace equation with the boundary conditions T=T1 and T=T2 at specified circles. The circles have radius R. We use the coordinate system, where the circles centers are located at points ( -d, 0) and (d, 0).

Consider the plane (x, y) as the complex plane z=(x,iy). Laplace equation is invariant with respect to conformal mappings, w=w(z). The idea is to apply a mapping such as the geometry of the problem becomes simple. The proper candidate is a bilinear mapping w = (a z- b) / (c z-d), which transforms circles into circles. With a proper choice of constants a, b, c, d, we can build a bilinear mapping, which transforms two above circles in the z-plane into two concentric circles in the w-plane with the centers at the coordinate origin.

To find the thermal flux through a contour L you need to integrate the value dL_x (dT/dy) - dL_y (dT/dx) over the contour. The function T is harmonic. Hence, it can be viewed as a real part of some complex function F. Let imaginary part of F is S. The functions T and S satisfy the Cauchy-Riemann conditions. Then
dL_x (dT/dy) - dL_y (dT/dx) = - dL_x (dS/dx) - dL_y (dS/dy).
The integrand is a full differential, and the flux is proportional to the difference S(B)-S(A), where B and A are points at the contour ends. [Generally speaking, the function S is multi-valued. When you go around a source and arrive again at the point A, the value S(B)-S(A) is not zero, because you jump on a different branch of S. To avoid this, one needs to make cuts in the complex plane.] When you apply a conformal mapping, then you contour is transformed into another contour, however the values of S at the contour ends remain the same. It means that the flux from one circle to another circle does not change under conformal mappings.

This observation makes life easier. Since we want to calculate the flux, but not the entire function T(x,y), then we do not need to determine w(z) explicitely. For our purpose, it is sufficient to use the property of bilinear mappings:

(w1-w2)(w3-w4) / ( (w1-w4)(w3-w2) ) =
(z1-z2)(z3-z4) / ( (z1-z4)(z3-z2) ) ,

where zj is mapped into wj.

Suppose we managed to choose a,b,c,d such as the left circle in the z-plane is mapped into the circle with the unit radius in the w-plane, and diametrically opposite points along the real axis in the z-plane , z4=(-d-R,0) and z3=(-d+R,0), are transformed into diametrically opposite points along the real axis in the w-plane, w4=(1,0) and w3=(-1,0). Apply the same mapping to the right circle, with similar requirements that z1 = (d+R,0) and z2=(d-R,0) are transformed into w1=(A,0) and w2=(-A,0). Substitute all this into the above property of bilinear mappings and get an equation for A. This is a quadratic equation, whose solutions are

A =( d/R + sqrt (d^2/R^2-1) )^P.

The power P is either 2 or -2. You can take any solution, it does not influence the final answer. Now you have a problem with two concentric circles. One circle has radius 1 and temperature T1. Another circle has radius A and temperature T2. A simple calculation gives the flux Q:

Q = 2*pi*k(T2-T1) / ln(A),

where k is the heat conductivity of the material. Substitute the numbers and get your answer.

Answer 2

I'm a little too lazy to solve this analytically. But since you've made no mention of the flow rates, I assume that the surface of the pipes remain at 10 and 50 C.

I've solved it using the finite element software, FEMLAB, and got ~ 14.35 W. Check out the figure below.

Some of the heat is transfered to the medium. In total 17.35 W of heat is transferred across the entire cylinder surface, however 14.35 W is transferred between the surfaces facing each other.

Note that if we had two walls instead of two cylinders, the heat transfer would simply be
0.1 * 16 * (50-10) /4 = 16 W
However the curvature of the cylindrical surfaces means that 1) there is a greater area for heat transfer, 2) the heat has to travel longer distances to get from one surface to the next.
So 14.35 W is realistic.

Answer 3

go to plumbing sections in Yahoo Answers

LMAO