Question

Consider steady one-dimensional conduction in a slab having a thickness of L and a constant thermal conductivity of k. The two ends are maintained at temperatures T_0 (at x = 0), and T_L (at x = L). There is a heat source, with strength A(x/L)(1 - x/L) W/cm^3. a) Define a set of dimensionless independent variables (depending on position), and a dimensionless dependent variable. b) Obtain a differential equation in which each term is dimensionless. c) Define an appropriate dimensionless parameter, and rewrite the differential equation obtained in b) as one in which each factor in each term is dimensionless.

Answer 1

The general 1 -D conduction equation is :

d/dx(KAdT/dx)=0 where A is the area and K is the constant

a) now thermal conductivity K= constant

then d/dx(AdT/dx)=0

or d^2T/dx^2+1/AdA/dxdT/dx=0

now heat source strength/heat Q=A(x/L)(1-x/L)

and we are considering heat trasfer through a slab therefore area A=constant

now heat flux from T0 to Tl is given by

$\dot{Q}$ =-KdT/dx=-K(Tl-To)/L

so $\dot{A(x/L)(1-x/L))}$ =-K(Tl-To)/L

that means properties of bar are independent variables and flow rate, conductivity are dependent variables.

b)now change of temperature with time can de defined by total change in energy in the system

so $\rho .V.Cp$ dT/dt= dQ/dt=d[A(x/L)(1-x/L)]/dt