Chemical Engineering 6453 Heat Transfer Prof. Geoff Silcox Spring 2005 Solution to Homework Assignment 6 Due Wednesday, 23 February, by 17:00.

Problem 1 A copper wire-coating operation is sketched below. The design and operating conditions have not been finalized and you have been asked to review and complete the design. The wire moves with uniform velocity V. In the first zone the wire is heated inductively in air over a length L1. The volumetric heating term is S, W/m3 and there are some convective losses given by h1. In the second zone, the wire passes through a fluidized bed of coating powder. In the third zone the coated wire is quenched in water before going to a cutting and spooling process. Given the following operating conditions and physical properties, calculate the temperature profile in the wire and specify all missing operating and design parameters. You may assume that the coating on the wire is thin and that it does not significantly affect the heat transfer process or dimensions of the wire. Include a copy of your code, output, and graphs as needed to support your one or two paragraphs of conclusions and recommendations.

V, T0

L1 L2

Coatedwire

Fluidizedbed of coatingpowder

h3, T∞3h1, T∞1

L3

Barewire

S

h2, T∞2

T(0) = T0 = 300 K h1 = 10 W/(m2 K) V = 0.50 m/s h2 = 1000 W/(m2 K) L1 = 0.20 m h3 = 5000 W/(m2 K) S = ? L3 = ? T(L1) = 425 K T(L1 + L2 + L3) = 300 K D = bare wire diameter = 1.5 mm T∞2 = 310 K L2 = 0.1 m T∞3 = 280 K T∞1 = 325 K

Solution The governing differential equation that applies to all portions of the process is

( ) 0dT PcV S h T Tdx A

ρ ∞− + + − = (1)

where P is the perimeter of the wire and A is its cross-sectional area. Note that S, h, and T∞ are functions of x. Equation (1) neglects conduction of heat in the direction of flow. It is solved in the MATLAB program, wireprc.m, using a node center grid and an inlet temperature of 300 K. The discretization scheme for (1) uses upwind differencing to ensure stability. The temperature at the end of the quench section is calculated using ( )1 20.25 9 8 3end n n nT T T T− −= − + (2) Equation (2) is based on the assumption that the temperature gradient in the x-direction is linear with x. The calculated profile that meets the required temperature specifications is shown in Figure 1. The source term for the heating section is 1.05x109 W/m3 and the length of the quench section, L3, is 0.24 m.

0 0.1 0.2 0.3 0.4 0.5 0.6300

320

340

360

380

400

420

440Temperature profile

Distance from inlet, m

T, K

elvi

n

Figure 1. Temperature profile for wire coating process. The source term for the heating section is 1.05x109 W/m3 and the length of the quench section, L3, is 0.24 m.