Transient Conduction in Cylinder:

Single Term Approximation

M.M. Yovanovich

DFCYL1TERM.MWS

One-dimensional transient conduction in long solid circular cylinder. Single term approximation.

The single term approximation is applicable for all values of the Biot number provided the

dimensionless time is greater than .

The single term approximations have the forms: and

where the Bessel-Fourier temperature and heat loss fraction coefficients are found from and . The first root of the characteristic

equation are obtained from the approximation and the parameter . For cooling the dimensionless temperature is defined as , and , , and .

> restart:

Dimensionless temperature.

> phi:= A[1]*exp(-delta[1]^2*Fo)*BesselJ(0,delta[1]*zeta);

Heat loss fraction.

> Q_Qi:= 1-B[1]*exp(-delta[1]^2*Fo);

Bessel-Fourier coefficients for temperature and heat loss fraction.

> A[1]:= 2*BesselJ(1,delta[1])/(delta[1]*(BesselJ(0,delta[1])^2 + BesselJ(1,delta[1])^2));

> B[1]:= 2*A[1]*BesselJ(1,delta[1])/delta[1];

First root of characteristic equation.

> delta[1]:= 2.4048255/(1+ (2.4048255/sqrt(2*Bi))^n)^(1/n);

> n:= 2.238;

Dimensionless system parameters.

> pars:= [Bi = h*a/k, Fo=alpha*t/a^2];

Example 1.

> case1:= (a=20/1000, k = 20, h = 100, alpha=20e-6, t = 6);

Calculation of system dimensionless parameters.

> parsvals:= subs(case1, pars);

Dimensionless temperature distribution.

> phi1:= evalf(subs(pars, case1, phi));

Calculation of heat loss fraction.

> Q_Qi1:= evalf(subs(pars, case1, Q_Qi));

> plot(phi1, zeta=0..1);

>