5.3. UNSTEADY-STATE HEAT CONDUCTION IN VARIOUS GEOMETRIES

5.3A. Introduction and Analytical Methods

In Section 5.2 we considered a simplified case of negligible internal resistance where the object has a very high thermal conductivity. Now we will consider the more general situation where the internal resistance is not small, and hence the temperature is not constant in the solid. The first case that we shall consider is one where the surface convective resistance is negligible compared to the internal resistance. This could occur because of a very large heat-transfer coefficient at the surface or because of a relatively large conductive resistance in the object.

To illustrate an analytical method of solving this first case, we will derive the equation for unsteady-state conduction in the x direction only in a flat plate of thickness 2H as shown in Fig. 5.3-1. The initial profile of the temperature in the plate at t = 0 is uniform at T = T0. At time t = 0, the ambient temperature is suddenly changed to T1 and held there. Since there is no convection resistance, the temperature of the surface is also held constant at T1. Since this is conduction in the x direction, Eq. (5.1-10) holds:

Equation 5.1-10


Figure 5.3-1. Unsteady-state conduction in a flat plate with negligible surface resistance.


The initial and boundary conditions are

Equation 5.3-1


Generally, it is convenient to define a dimensionless temperature Y so that it varies between 0 and 1. Hence,

Equation 5.3-2


Substituting Eq. (5.3-2) into (5.1-10),

Equation 5.3-3


Redefining the boundary and initial conditions,

Equation 5.3-4


A convenient procedure to use to solve Eq. (5.3-3) is the method of separation of variables, which leads to a product solution

Equation 5.3-5


where A and B are constants and a is a parameter. Applying the boundary and initial conditions of Eq. (5.3-4) to solve for these constants in Eq. (5.3-5), the final solution is an infinite Fourier series (G1):

Equation 5.3-6


Hence from Eq. (5.3-6), the temperature T at any position x and time t can be determined. However, these types of equations are very time-consuming to use, and convenient charts have been prepared, which are discussed in Sections 5.3B, 5.3C, 5.3D, and 5.3E, where a surface resistance is present.

5.3B. Unsteady-State Conduction in a Semi-infinite Solid

In Fig. 5.3-2 a semi-infinite solid is shown that extends to ∞ in the +x direction. Heat conduction occurs only in the x direction. Originally, the temperature in the solid is uniform at T0. At time t = 0, the solid is suddenly exposed to or immersed in a large mass of ambient fluid at temperature T1, which is constant. The convection coefficient h in W/m2 · K or btu/h · ft2 · °F is present and is constant; that is, a surface resistance is present. Hence, the temperature TS at the surface is not the same as T1.

Figure 5.3-2. Unsteady-state conduction in a semi-infinite solid.


The solution of Eq. (5.1-10) for these conditions has been obtained (S1) and is

Equation 5.3-7


where x is the distance into the solid from the surface in m (SI units), t = time in s, α = k/ρcp in m2/s. In English units, x = ft, t = h, and α = ft2/h. The function erfc is (1 − erf), where erf is the error function and numerical values are tabulated in standard tables and texts (G1, P1, S1); Y is fraction of unaccomplished change (T1T)/(T1T0); and 1 − Y is fraction of change.

Figure 5.3-3, calculated using Eq. (5.3-7), is a convenient plot used for unsteady-state heat conduction into a semi-infinite solid with surface convection. If conduction into the solid is slow enough or h is very large, the top line with is used.

Figure 5.3-3. Unsteady-state heat conducted in a semi-infinite solid with surface convection. Calculated from Eq. (5.3-7)(S1).


EXAMPLE 5.3-1. Freezing Temperature in the Ground

The depth in the soil of the earth at which freezing temperatures penetrate is often of importance in agriculture and construction. On a certain fall day, the temperature in the earth is constant at 15.6°C (60°F) to a depth of several meters. A cold wave suddenly reduces the air temperature from 15.6 to −17.8°C (0°F). The convective coefficient above the soil is 11.36 W/m2 · K (2 btu/h · ft2 · °F). The soil properties can be assumed as α = 4.65 × 107 m2/s (0.018 ft2/h) and k = 0.865 W/m · K (0.5 btu/h · ft · °F). Neglect any latent heat effects. Use SI and English units.

  1. What is the surface temperature after 5 h?

  2. To what depth in the soil will the freezing temperature of 0°C (32°F) penetrate in 5 h?

Solution: This is a case of unsteady-state conduction in a semi-infinite solid. For part (a), the value of x which is the distance from the surface is x = 0 m. Then the value of is calculated as follows for t = 5 h, α = 4.65 × 107 m2/s, k = 0.865 W/m · °C, and h = 11.36 W/m2 · °C. Using SI and English units,


Also,


Using Fig. 5.3-3, for = 0 and , the value of 1 − Y = 0.63 is read off the curve. Converting temperatures to K, T0 = 15.6°C + 273.2 = 288.8 K (60°F) and T1 = −17.8°C + 273.2 = 255.4 K (0°F). Then


Solving for T at the surface after 5 h,


For part (b), T = 273.2 K or 0°C, and the distance x is unknown. Substituting the known values,


From Fig. 5.3-3 for (TT0)/(T1T0) = 0.467 and , a value of 0.16 is read off the curve for . Hence,


Solving for x, the distance the freezing temperature penetrates in 5 h,



5.3C. Unsteady-State Conduction in a Large Flat Plate

A geometry that often occurs in heat-conduction problems is a flat plate of thickness 2x1 in the x direction and having large or infinite dimensions in the y and z directions, as shown in Fig. 5.3-4. Heat is being conducted only from the two flat and parallel surfaces in the x direction. The original uniform temperature of the plate is T0; at time t = 0, the solid is exposed to an environment at temperature T1 and unsteady-state conduction occurs. A surface resistance is present.

Figure 5.3-4. Unsteady-state conduction in a large flat plate.


The numerical results of this case are presented graphically in Figs. 5.3-5 and 5.3-6. Figure 5.3-5, from Gurney and Lurie (G2), is a convenient chart for determining the temperatures at any position in the plate and at any time t. The dimensionless parameters used in this and subsequent unsteady-state charts in this section are given in Table 5.3-1 (x is the distance from the center of the flat plate, cylinder, or sphere; x1 is one-half the thickness of the flat plate, radius of cylinder, or radius of sphere; x = distance from the surface for a semi-infinite solid.)

Table 5.3-1. Dimensionless Parameters for Use in Unsteady-State Conduction Charts
 
SI units: α = m2/s, T = K, t = s, x = m, x1 = m, k = W/m · K, h = W/m2 · K
English units: α = ft2/h, T = °F, t = h, x = ft, x1 = ft, k = btu/h · ft · °F, h = btu/h · ft2 · °F
Cgs units: α = cm2/s, T = °C, t = s, x = cm, x1 = cm, k = cal/s · cm · °C, h = cal/s · cm2 · °C

Figure 5.3-5. Unsteady-state heat conduction in a large flat plate. [From H. P. Gurney and J. Lurie, Ind. Eng. Chem., 15, 1170 (1923).]


Figure 5.3-6. Chart for determining temperature at the center of a large flat plate for unsteady-state heat conduction. [From H. P. Heisler, Trans. A.S.M.E., 69, 227 (1947). With permission.]


When n = 0, the position is at the center of the plate in Fig. 5.3-5. Often the temperature history at the center of the plate is quite important. A more accurate chart for determining only the center temperature is given in Fig. 5.3-6, the Heisler (H1) chart. Heisler (H1) has also prepared multiple charts for determining the temperatures at other positions.

EXAMPLE 5.3-2. Heat Conduction in a Slab of Butter

A rectangular slab of butter which is 46.2 mm thick at a temperature of 277.6 K (4.4°C) in a cooler is removed and placed in an environment at 297.1 K (23.9°C). The sides and bottom of the butter container can be considered to be insulated by the container side walls. The flat top surface of the butter is exposed to the environment. The convective coefficient is constant at 8.52 W/m2 · K. Calculate the temperature in the butter at the surface, at 25.4 mm below the surface, and at 46.2 mm below the surface at the insulated bottom after 5 h of exposure.

Solution: The butter can be considered as a large, flat plate with conduction vertically in the x direction. Since heat is entering only at the top face and the bottom face is insulated, the 46.2 mm of butter is equivalent to a half plate with thickness x1 = 46.2 mm. In a plate with two exposed surfaces, as in Fig. 5.3-4, the center at x = 0 acts as an insulated surface, and both halves are mirror images of each other.

The physical properties of butter from Appendix A.4 are k = 0.197 W/m · K, cp = 2.30 kJ/kg · K = 2300 J/kg · K, and ρ = 998 kg/m3. The thermal diffusivity is


Also, x1 = 46.2/1000 = 0.0462 m.

The parameters needed for use in Fig. 5.3-5 are


For the top surface, where x = x1 = 0.0462 m,


Then, using Fig. 5.3-5,


Solving, T = 292.2 K (19.0°C).

At the point 25.4 mm from the top surface, or 20.8 mm from the center, x = 0.0208 m, and


From Fig. 5.3-5,


Solving, T = 288.3 K (15.1°C).

For the bottom point, or 0.0462 m from the top, x = 0 and


Then, from Fig. 5.3-5,


Solving, T = 287.4 K (14.2°C). Alternatively, using Fig. 5.3-6, which is only for the center point, Y = 0.53 and T = 286.8 K (13.6°C).


5.3D. Unsteady-State Conduction in a Long Cylinder

Here we consider unsteady-state conduction in a long cylinder where conduction occurs only in the radial direction. The cylinder is long so that either conduction at the ends can be neglected or the ends are insulated. Charts for this case are presented in Fig. 5.3-7 for determining the temperatures at any position and Fig. 5.3-8 for the center temperature only.

Figure 5.3-7. Unsteady-state heat conduction in a long cylinder. [From H. P. Gurney and J. Lurie, Ind. Eng. Chem., 15, 1170 (1923).]


Figure 5.3-8. Chart for determining temperature at the center of a long cylinder for unsteady-state heat conduction. [From H. P. Heisler, Trans. A.S.M.E., 69, 227 (1947). With permission.]


EXAMPLE 5.3-3. Transient Heat Conduction in a Can of Pea Purée

A cylindrical can of pea purée (C2) has a diameter of 68.1 mm and a height of 101.6 mm and is initially at a uniform temperature of 29.4°C. The cans are stacked vertically in a retort, and steam at 115.6°C is admitted. For a heating time of 0.75 h at 115.6°C, calculate the temperature at the center of the can. Assume that the can is in the center of a vertical stack of cans and that it is insulated on its two ends by the other cans. The heat capacity of the metal wall of the can will be neglected. The heat-transfer coefficient of the steam is estimated as 4540 W/m2 · K. Physical properties of purée are k = 0.830 W/m · K and α = 2.007 × 107 m2/s.

Solution: Since the can is insulated at the two ends, we can consider it as a long cylinder. The radius is x1 = 0.0681/2 = 0.03405 m. For the center with x = 0,


Also,


Using Fig. 5.3-8 from Heisler for the center temperature,


Solving, T = 104.4°C.


5.3E. Unsteady-State Conduction in a Sphere

Figure 5.3-9 shows a chart by Gurney and Lurie for determining the temperatures at any position in a sphere. Figure 5.3-10 is a chart by Heisler for determining only the center temperature in a sphere.

Figure 5.3-9. Unsteady-state heat conduction in a sphere. [From H. P. Gurney and J. Lurie, Ind. Eng. Chem., 15, 1170 (1923).]


Figure 5.3-10. Chart for determining the temperature at the center of a sphere for unsteady-state heat conduction [From H. P. Heisler, Trans. A.S.M.E., 69, 227 (1947). With Permission.]


5.3F. Unsteady-State Conduction in Two- and Three-Dimensional Systems

The heat-conduction problems considered so far have been limited to one dimension. However, many practical problems involve simultaneous unsteady-state conduction in two and three directions. We shall illustrate how to combine one-dimensional solutions to yield solutions for several-dimensional systems.

Newman (N1) used the principle of superposition and showed mathematically how to combine the solutions for one-dimensional heat conduction in the x, the y, and the z direction into an overall solution for simultaneous conduction in all three directions. For example, a rectangular block with dimensions 2x1, 2y1, and 2z1 is shown in Fig. 5.3-11. For the Y value in the x direction, as before,

Equation 5.3-8


Figure 5.3-11. Unsteady-state conduction in three directions in a rectangular block.


where Tx is the temperature at time t and position x distance from the center line, as before. Also, n = x/x1, m = k/hx1, and , as before. Then for the y direction,

Equation 5.3-9


and n = y/y1, m = k/hy1, and . Similarly, for the z direction,

Equation 5.3-10


Then, for simultaneous transfer in all three directions,

Equation 5.3-11


where Tx,y,z is the temperature at the point x, y, z from the center of the rectangular block. The value of Yx for the two parallel faces is obtained from Figs. 5.3-5 and 5.3-6 for conduction in a flat plate. The values of Yy and Yz are similarly obtained from the same charts.

For a short cylinder with radius x1 and length 2y1, the following procedure is followed. First Yx for the radical conduction is obtained from the figures for a long cylinder. Then Yy for conduction between two parallel planes is obtained from Fig. 5.3-5 or 5.3-6 for conduction in a flat plate. Then,

Equation 5.3-12


EXAMPLE 5.3-4. Two-Dimensional Conduction in a Short Cylinder

Repeat Example 5.3-3 for transient conduction in a can of pea purée but assume that conduction also occurs from the two flat ends.

Solution: The can, which has a diameter of 68.1 mm and a height of 101.6 mm, is shown in Fig. 5.3-12. The given values from Example 5.3-3 are x1 = 0.03405 m, y1 = 0.1016/2 = 0.0508 m, k = 0.830 W/m · K, α = 2.007 × 107 m2/s, h = 4540 W/m2 · K, and t = 0.75(3600) = 2700 s.

Figure 5.3-12. Two dimensional conduction in a short cylinder in Example 5.3-4.


For conduction in the x (radial) direction as calculated previously,


From Fig. 5.3-8 for the center temperature,


For conduction in the y (axial) direction for the center temperature,


Using Fig. 5.3-6 for the center of a large plate (two parallel opposed planes),


Substituting into Eq. (5.3-12),


Then,


This compares with 104.4°C obtained in Example 5.3-3 for only radial conduction.


5.3G. Charts for Average Temperature in a Plate, Cylinder, and Sphere with Negligible Surface Resistance

If the surface resistance is negligible, the curves given in Fig. 5.3-13 will give the total fraction of unaccomplished change, E, for slabs, cylinders, or spheres for unsteady-state conduction. The value of E is

Equation 5.3-13


Figure 5.3-13. Unsteady-state conduction and average temperatures for negligible surface resistance. (From R. E. Treybal, Mass Transfer Operations, 2nd ed. New York: McGraw-Hill Book Company, 1968. With permission.)


where T0 is the original uniform temperature, T1 is the temperature of the environment to which the solid is suddenly subjected, and Tav is the average temperature of the solid after t hours.

The values of Ea, Eb, and Ec are each used for conduction between a pair of parallel faces, as in a plate. For example, for conduction in the a and b directions in a rectangular bar,

Equation 5.3-14


For conduction from all three sets of faces,

Equation 5.3-15


For conduction in a short cylinder 2c long and radius a,

Equation 5.3-16


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