Transient Conduction of Heat in Solids | Thermal Engineering

In this article we will discuss about transient conduction of heat in solids with infinite thermal conductivity and finite conduction.

Transient Conduction of Heat in Solids with Infinite Thermal Conductivity K → ∞ (Lumped Parameter Analysis):

Solutions to the many of the transient heat flow problems are obtained by the lumped parameter analysis which presumes that the solid possesses infinitely large thermal conductivity. Internal conduction resistance is then so small that heat flow to or from the solid is controlled primarily by the convective resistance.

Temperature gradients are negligible within the solid. Consequently the solid is space wise isothermal t ≠ f (x, y, z) with temperature varying only with the time t = f(τ). Temperature, though changing with time, is nevertheless uniform throughout the solid at any time.

Typical examples of this type of heat flow are:

(i) Cooling of a small metal casting or a billet in quenching bath after its removal from the furnace

(ii) Heating or cooling of a fine thermocouple wire due to change in ambient temperature

Fig. 6.1. shows a general lump of material comprising the system of interest. A body of surface area A, volume V, density p, thermal conductivity k, specific heat c and initial temperature t_i has been exposed to the surroundings maintained at temperature t_a. The transient response of the solid can be determined by relating its rate of change of internal energy with convective heat exchange at the surface.

That is-

This expression can be rearranged and integrated; temperature t and the time τ are the two variables.

The integration constant C₁ is evaluated from the initial conditions: t = t_i at τ = 0; t_i symbolizes the body temperature at the commencement of the cooling or heating process. Therefore C₁ = log_e (t_i – t_a) and hence-

Following points are worth noting:

1. The body temperature falls or rises exponentially with time and the rate depends on the parameter (hA/ρVc). Theoretically the body takes infinite time to approach the temperature of surroundings and thus attain the steady state conditions. However the difference between t and t_a becomes extremely small after a short time and beyond that period the body temperature becomes practically equal to the ambient temperature.

2. The quantity (ρVc/hA) has the dimensions of time and is called the thermal time constant. Its value is indicative of the rate of response of a system to a sudden change in the environmental temperature; how fast a body will respond to a change in the environmental temperature.

3. The dimensionless argument of the exponential can be arranged in different forms such as-

α = [k/ρc] is the thermal diffusivity of the solid, and I is a characteristic length equal to the ratio of the volume of the solid to its surface area.

4. For simple geometrical shapes, the values of characteristic length l are-

For a flat plate (thickness δ, breadth b, and height h) the heat exchange occurs from both the sides; area exposed for heat transfer is 2bh.

The characteristics length then equals:

l = (δbh/2bh) = δ/2, i.e., half the plate thickness.

i. The non-dimensional factor (α τ/l²) is called the Fourier number, F₀. It signifies the degree of penetration of heating or cooling effect through a solid. For instance, a large time τ would be required to obtain a significant temperature change for small values of (α/l²).

ii. The non-dimensional factor (hl/k) is called the Biot number, B_i. It gives an indication of the ratio of internal (conduction) resistance to the surface (convection) resistance. A small value of B_; implies that the system has a small conduction (internal) resistance, i.e., relatively small temperature gradient or the existence of a practically uniform temperature within the system. The convective resistance then predominates and the convective heat exchange controls the transient phenomenon. Essentially this has been the basic assumption in the lumped parameter analysis made above.

The application of the lumped parameter approach to bodies with shapes similar to plates, cylinders or spheres does indicate that the temperatures in the body differ by less than 5% at any time for a value of B_i < 0.1. Small Biot numbers can hold with thin plates, and with large thermal conductivity k and small heat transfer coefficient h.

The lumped parameter solution for transient conduction can be conveniently stated as-

Instantaneous and Total Heat Flow Rate:

The instantaneous heat flow rate Q_; may be computed as follows:

and the total heat flow (loss or gain) is obtained by integrating equation 6.5 over the time interval τ = 0 to τ = τ.

Example 1:

An iron (k = 65 W/mK) billet measuring 20 x 15 x 80 cm is exposed to a convective flow resulting in convection coefficient h = 11.5 W/m²K. Determine the Biot number and the suitability of a lumped analysis to represent the cooling rate if the billet is initially hotter than the environment.

Solution:

The characteristic linear dimension defined as the ratio of the volume of the billet to its surface area works out to be-

Since the Biot number is less than 0.1, the internal temperature gradients are small. Consideration of the billet as a lumped system would be quite accurate; it will introduce an error of no more than 5%.

Transient Conduction of Heat in Solids with Finite Conduction and Convective Resistance (0 < B_i < 100):

Consider the heating or cooling of a plane wall of thickness l = 2δ and extending to infinity in the y and z directions. Initially the wall is at uniform temperature t_i, and suddenly both surfaces (x = ± δ) are exposed to and maintained at the ambient temperature t_a.

The controlling differential equation for the transient heat conduction is:

d²t/dx² = 1 dt/α dτ

and the appropriate boundary conditions are:

(i) t = t_i at τ = 0; initially the wall is at uniform temperature t_i

(ii) dt/dx = 0 at x = 0; symmetrical nature of the temperature profile within the plane wall; symmetry in conduction occurs at the mid plane (x = 0) of the wall.

(iii) kA (dt/dx) = hA (t – t_a) at x = ± δ. This condition stems from the fact that conduction heat transfer equals the convective heat transfer at the wall surface.

The solution of the controlling differential equation in conjunction with initial boundary conditions would give an expression for temperature variation both with time and position. The solutions obtained after rigorous mathematical analysis indicate that-

Obviously when conduction resistance is not negligible, the temperature history becomes a function of Biot number hl/k, Fourier number ατ/l² and the dimensionless parameter x/l which indicates the location of point within the plate where temperature is to be obtained. In case of cylinders and spheres x/l is replaced by r/R.

Graphical charts have been prepared for the equation 6.10 in a variety of forms. The Heisler charts given in Figs. 6.5 to 6.7 depict the dimensionless temperature (t₀ – t_a)/(t_i – t_a) versus F₀ for various values of 1/B_i for solids of different geometrical shapes such as plates, cylinders and spheres.

These charts give the temperature history of the solid at its mid plane, x = 0. Temperatures at other locations are worked out by multiplying the mid-plane temperature by the correction factors read from charts given in Fig. 6.8 to 6.10.

Use is made of the following relationship:

The values of Biot number and Fourier number, as used in the Heisler charts, are evaluated on the basis of a characteristic parameter s which is the semi-thickness in case of plates and the surface radius in case of cylinders and spheres.

The Heisler charts are extensively used to determine the temperature distribution and heat flow rate when both conduction and convection resistances are almost of equal importance.

Example 2:

A large steel plate 50 mm thick is initially at a uniform temperature of 425°C. It is suddenly exposed on both sides to an environment with convective coefficient 285 W/m²K and temperature 65°C. Determine the centre line temperature and the temperature inside the plate 12.5 mm from the mid plane after 3 minutes.

For Steel:

Thermal conductivity k = 42.5 W/mK

Thermal diffusivity α = 0.043 m²/hr

Solution:

The characteristic linear dimension for a flat plate equals half the plate thickness, i.e.,

Since Biot number is greater than 0.1, the internal temperature gradients are not small and so the internal resistance cannot be neglected. Consideration of the plate as a lumped system would then be inappropriate. Further, the Biot number is less than 100, and accordingly the transient solution can be obtained by employing Heisler charts.

Example 3:

The nose section of a missile is formed of a 6 mm thick stainless steel plate and is held initially at a uniform temperature of 88°C. The missile enters the denser layers of the atmosphere at a very high velocity. The effective temperature of air surrounding the nose region attains the value 2200°C and the surface convective coefficient is estimated at 3405 W/m²K. Make calculations for the maximum permissible time in these surroundings if the maximum metal temperature is not to exceed 1095°C. Also work out the inside surface temperature under these conditions.

The constant values for steel properties are: density ρ = 7800 kg/m³; specific heat c = 465 J/kgK and thermal conductivity k = 54 W/mK.

Solution:

The nose section may be idealised as a flat plate for which the characteristic linear dimension equals half the plate thickness, i.e., l = 6/2 = 3 mm

Since Biot number is greater than 0.1, the lumped analysis would be inappropriate. Further B_i < 100 and accordingly the transient solution can be obtained by employing Heisler charts.

For the flat plate, the following parametric values apply: