Consider a plane wall of thickness 2L , a long cylinder of radius r_{o}, and a sphere of radius r_{o} initially at a uniform temperature T_{i}, as shown in Figure 4.8.
Figure 4.8 Schematic of the simple geometries in which heat transfer is one dimensional
At time t=0, each geometry is placed in a large medium that is at a constant temperature and kept in that medium for t>0. Heat transfer takes place between these bodies and their environments by convection with a uniform and constant heat transfer coefficient h . Note that all three cases possess geometric and thermal symmetry: the plane wall is symmetric about its center plane (x=0), the cylinder is symmetric about its centerline (r=0), and the sphere is symmetric about its center point (r=0). We neglect radiation heat transfer between these bodies and their surrounding surfaces, or incorporate the radiation effect into the convection heat transfer coefficient h .
The variation of the temperature profile with time in the plane wall is illustrated in Figure 4.9. When the wall is first exposed to the surrounding medium at >T_{i}at t=0 , the entire wall is at its initial temperature T_{i}. But the wall temperature at and near the surfaces starts to drop as a result of heat transfer from the wall to the surrounding medium. This creates a temperature gradient in the wall and initiates heat conduction from the inner parts of the wall toward its outer surfaces. Note that the temperature at the center of the wall remains at T_{i} until t= t_{2}, and that the temperature profile within the wall remains symmetric at all times about the center plane. The temperature profile gets flatter and flatter as time passes as a results of heat transfer, and eventually becomes uniform at T= . That is, the wall reaches thermal equilibrium with its surroundings. At that point, the heat transfer stops since there is no longer a temperature difference. Similar discussions can be given for the long cylinder or sphere.
Figure 4.9 Transient temperature profiles in a plane wall exposed to convection from its surfaces for T_{i}_{}> .
The formulation of the problems for the determination of the one dimensional transient temperature distribution T(x,t) in a wall results in a partial differential equation, which can be solved using advanced mathematical techniques. The solution, however, normally involves infinite series, which are inconvenient and time consuming to evaluate. Therefore, there is a clear motivation to present the solution in tabular or graphical form. However, the solution involves the parameters x,L,t,k,a,h,T_{i}, and , which are too many to make any graphical presentation of the results practical. In order to reduce the number of parameters, we nondimensionalize the problem by defining the following dimensionless quantities:
Dimensionless Temperature: 

Dimensionless distance from the center: 

Dimensionless heat transfer coefficient: 
(Biot number) 
Dimensionless time: 
(Fouriernumber)

The nondimensionalization enables us to present the temperature in terms of three parameters only: X, Bi and t . This makes it practical to present the solution in the graphical form. The dimensionless quantities defined above for a plane wall can also be used for a cylinder or sphere by replacing the space variable x by r and the half thickness L by the outer radius r_{o}. Note that the characteristic length in the definition of the Biot number is taken to be the halfthickness L of the plane wall, and the radius r_{o} for the long cylinder and sphere instead of V/A used in lumped system analysis.
The one dimensional transient heat conduction problem described above can be solved exactly for any of three geometries, but the solution involves infinite series, which are difficult to deal with. However, the terms in the solutions converge rapidly with increasing time, and for t > 0.2, keeping the first term and neglecting all the remaining terms in the series results in an error under 2 %. We are usually interested in the solution for time with t > 0.2, and thus it is very convenient to express the solution using this one term approximation, given as
Plane wall:
Once the Bi number is known, the above relations can be used to determine he temperature anywhere in the medium. The determination of the constants A_{1} and l_{1}usually requires interpolation. For those who prefer reading charts to interpolating, the relations above are plotted and the oneterm approximation solutions are presented in graphical form, known as the transient temperature charts. Note that the charts are sometimes difficult to read, and they are subjected to reading errors. Therefore, the relations above should be preferred to the charts.
Table 4.1 Coefficients used in the oneterm approximate solution of transient onedimensional heat conduction in plane walls, cylinders and spheres ( Bi = hL/k for plane wall of thickness 2L , and Bi=hr_{o}/k for a cylinder or sphere of radius r_{o})

Plane wall 
Cylinder 
Sphere 
Bi 

A_{1} 

A_{1} 

A_{1} 

0.0998 






0.1410 






0.1987 






0.2425 






0.2791 






0.3111 






0.4328 






0.5218 






0.5932 






0.6533 






0.7051 






0.7506 






0.7910 






0.8274 






0.8603 






1.0769 






1.1925 






1.2646 






1.3138 






1.3496 






1.3766 






1.3978 






1.4149 






1.4289 






1.4961 






1.5202 






1.5325 






1.5400 






1.5552 






1.5708 





Table 4.2 The zeroth and first order Bessel functions of the first kind



0.0 
1.0000 
0.0000 
0.1 
0.9975 
0.0499 
0.2 
0.9900 
0.0995 
0.3 
0.9776 
0.1483 
0.4 
0.9604 
0.1960 
0.5 
0.9385 
0.2423 
0.6 
0.9120 
0.2867 
0.7 
0.8812 
0.3290 
0.8 
0.8463 
0.3688 
0.9 
0.8075 
0.4059 
1.0 
0.7652 
0.4400 
1.1 
0.7196 
0.4709 
1.2 
0.6711 
0.4983 
1.3 
0.6201 
0.5220 
1.4 
0.5669 
0.5419 
1.5 
0.5118 
0.5579 
1.6 
0.4554 
0.5699 
1.7 
0.3980 
0.5778 
1.8 
0.3400 
0.5815 
1.9 
0.2818 
0.5812 
2.0 
0.2239 
0.5767 
2.1 
0.1666 
0.5683 
2.2 
0.1104 
0.5560 
2.3 
0.0555 
0.5399 
2.4 
0.0025 
0.5202 
2.6 
0.0968 
0.4708 
2.8 
0.1850 
0.4097 
3.0 
0.2601 
0.3391 
3.2 
0.3202 
0.2613 
The transient temperature charts in Figures 4.10, 4.11 and 4.12 for a large plane wall, long cylinder and sphere were presented by M.P.Heisler in 1947 and are called Heisler charts. They were supplemented in 1961 with transient heat transfer charts by H.Grober. There are three charts associated with each geometry: the first chart is to determine the temperature T_{o} at the center of the geometry at a given time t. The second chart is to determine the temperature at other locations at the same time in terms of T_{o}. The third chart is to determine the total amount of heat transfer up to the time t. These plots are valid for t > 0.2.
Fig. 4.10 Transient Temperature and Heat Transfer Charts for a Plane Wall of thickness 2L initially at a uniform temperature T_{i} subjected to convection from both sides to an environment at temperature with a convection coefficient of h
Fig. 4.11 Transient Temperature and Heat Transfer Charts for a long cyinder of radius r_{o}initially at a uniform temperature subjected to convection from all sides to an environment at temperature with a convection coefficient of h
Fig. 4.12 Transient Temperature and Heat Transfer Charts for a sphere of radius r_{o}initially at a uniform temperature T_{i} subjected to convection from all sides to an environment at temperature with a convection coefficient of h
Note that the case 1/Bi=k/hL=0 corresponds to , which corresponds to the case of specified surface temperature . That is, the case in which the surfaces of the body are suddenly brought to the temperature at t=0 and kept at at all times can be handled by setting h to infinity (Fig. 4.13).
The temperature of the body changes from the initial temperature T_{i} to the temperature of the surroundings at the end of the transient heat conduction process. Thus, the maximum amount of heat that a body gain (or lose if T_{i}> ) is simply the change in the energy content of the body. That is,
(kJ) 
(4.19) 
Where m is the mass, V is the volume, r is the density, and C_{p} is the specific heat of the body. Thus, Q_{max} represents the amount of heat transfer for . The amount of heat transfer Q at a finite time t will obviously be less than this maximum. The ratio Q/Q_{max} is plotted in Figures 4.10c, 4.11c, and 4.12 c against the variables Bi and for the large plane wall, long cylinder and sphere, respectively. Note that once the fraction of heat transfer Q/Q_{max} has been determined from these charts for the given t , the actual amount of heat transfer by that time can be evaluated by multiplying this fraction by Q_{max}. A negative sign for Q_{max} indicates that heat is leaving the body (Fig. 4.14).
FIGURE 4.13 The Specified surface temperature corresponds to the case of convection to an environment at with a convection coefficient h that is infinite
Figure 4.14 The fraction of total heat transfer Q/Q_{max}up to a specified time t is determined using the Grober charts
The fraction of heat transfer can also be determined from the following relations, which are based on the oneterm approximations discussed above:
Plane wall

(4.20) 
Cylinder

(4.21) 
Sphere

(4.22) 
The use of the Heisler/Grober charts and the oneterm solutions discussed above is limited to the conditions specified at the beginning of this section: the body is initially at a uniform temperature, the temperature of the medium surrounding the body and the convection heat transfer coefficient are constant and uniform, and there is no energy generation in the body.
We discussed the physical significance of the Biot number earlier and indicated that it is a measure of the relative magnitude of the two heat transfer mechanisms: convection at the surface and conduction through the solid. A small value of Bi indicates that the inner resistance of the body to the heat conduction is small relative to the resistance to convection between the surface and the fluid. As a result, the temperature distribution within the solid becomes fairly uniform, and lumped system analysis becomes applicable. Recall that when Bi<0.1 , the error in assuming the temperature within the body to be uniform is negligible.
To understand the physical significance of the Fourier number , we express it as (Figure 4.15.

(4.23) 
Therefore, the Fourier number is a measure of heat conducted through a body relative to heat stored. Thus, a large value of the Fourier number indicates faster propagation of heat through a body.
Figure 4.15 Fourier number at time t can be viewed as the ratio of the rate of heat conducted to the rate of heat stored at that time
Perhaps you are wondering about what constitutes an infinitely large plate or an infinitely long cylinder. After all, nothing in this world in infinite. A plate whose thickness is small relative to the other dimensions can be modeled as an infinitely large plate, except very near the outer edges. But the edge effects on large bodies are usually negligible, and thus a large plane wall such as the wall of a house can be modeled as an infinitely large wall for heat transfer purposes. Similarly, a long cylinder whose diameter is small relative to its length can be analyzed as an infinitely long cylinder. Thus use of the transient temperature charts and the oneterm solutions is illustrated in the following examples.