# One-dimensional transient heat conduction in cylinder

### From Thermal-FluidsPedia

A long cylinder with radius of *r*_{o} and a uniform initial temperature of *T*_{i} is exposed to a fluid with temperature of (). The convective heat transfer coefficient between the fluid and cylinder is *h*. Assuming that there is no internal heat generation and constant thermophysical properties, obtain the transient temperature distribution in the cylinder.

Since the temperature changes along the r-direction only, the energy equation is

subject to the following boundary and initial conditions

Defining the following dimensionless variables

eqs. (1) – (4) will be nondimensionalized as

Assuming that the temperature can be expressed as

and substituting eq. (10) into eq. (6), one obtains

which can be rewritten as the following two equations

Equation is a Bessel’s equation of zero order and has the following general solution

where *J*_{0} and *Y*_{0} are Bessel functions of the first and second kind, respectively. The general solution of eq. (13) is

where *C*_{1},*C*_{2}, and *C*_{3} are integral constants.
The boundary conditions for eq. (12) can be obtained by substituting eq. (10) into eqs. (7) and (8), i.e.,

The derivative of Θ is

Since , *C*_{2} must be zero. Substituting eqs. (14) and (18) into eq. (17) and considering *C*_{2} = 0, we have

where *n* is an integer. The eigenvalue λ_{n} can be obtained by solving eq. (19) using an iterative procedure. The dimensionless temperature with eigenvalue λ_{n} is

where *C*_{n} = *C*_{1}*C*_{3}. Equation (20) is a solution that satisfies eqs. (6) – (8) but not eq. (9). For a linear problem, the sum of different θ_{n} for each value of *n* also satisfies eqs. (6) – (8).

Substituting eq. (21) into eq. (9) yields

Multiplying the above equation by and integrating the resulting equation in the interval of (0, 1), one obtains

According to the orthogonal property of Bessel’s function, the integral on the right-hand side equals zero if but it is not zero if *m* = *n*. Therefore, we have

Changing notation from *m* to *n*, we get

thus, the dimensionless temperature becomes

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, *Advanced Heat and Mass Transfer*, Global Digital Press, Columbia, MO.