Introduction:- If the temperature of the body does not very with time it said to be in steady state. if there is an abrupt change in its surface temperature it attains an equilibrium temp. or a steady state after a some time,during this period the temperature varies with the time and the body said to be in an unsteady or transient state. Conduction of heat is unstady state refers to the transient condition wherein the heat flow and the temperature distribution at any point of the system vary contionusly with time.

Example:- Cooling of ic engine; Automobile engines; Heating and cooling of metal billets; Cooling and freezing of food; Heat treatment of metals; Brick burning Vulcanization of rubber

Non periodic variation:- In this system, the temperature at any point varies non-linearly with time. Example:- 1. heating of an ingot in a furnace. 2. Cooling of bars,metal billets.

Periodic variation:- In this system temperature undergo periodic changes which are either regular or irregular but definetly cyclic. Example:- 1. Cylinder of an I.C. engine; 2. Building during a period of 24 hours; 3. Surface of earth during 24 hours; 4. Heat process of regenerators ;

All solids have a finite thermal conductivity and there will be always a temperature gradient inside the solids whenever heat is added or removed. However,for solids of large thermal conductivity with surface areas that are large in proportion to their volume likes plates and thin metallic wire,the internal resistence can be assmed to be small or negligible in comparison with the covective resistance at the surface. example: 1. Heat treatment of metals; 2. Time response of thermocouple and thermometer;

The process in which the internal resistance is assumed negligible in comparison with its surface resistance is called Newtonian heating or cooling process. The temperature,in this process is considered to be uniform at a given time. Such analysis is called Lumped Parameter Analysis. let us consider a body whose initial temperature is and it is placed in ambient air or any liquid at a constant temperature. The transient response of the body can be determined by relating its rate of internal energy with convective exchange at the surface.

=density of solid =volume of the body =specific heat of of body =unit surface conductance =temperature of body at any time =surface area of the body =ambient temperature =time

After re arranging the equation, The boundary condition are ; At Hence = =0

Eqn.gives the temperature distribution in the body for newnotian heating or cooling and it indicates that temperature rises exponentially with time as shown in the figure.

The quantity has the dimensions of time and is called thermal time constant.denoted by its value is indicative of the the rate of response to a change in its environmental temperature. where Resistance to convective heat transfer Lumped thermal capacitance of solid

Fig shows that any increase in Rth and Cth will cause a solid to respond more slowly to changes in its thermal environmental and will increase the time required to attain the thermal equilibrium. The power on exponential can be arranged in such form:- Where Characteristic length=

The values of characteristic length for simple geometric shapes are given below; FLATE PLATE: CYLINDER: SPHERE: CUBE:

The non – dimensions factor is called the biot number. It gives an indication of the ratio of internal resistance to surface resistance.when the value of biot number small,it indicates that the system has a small internal resistance. If the biot number is less than 0.1 then lumped heat capacity approch can be used to advantage with simple shapes as plates,cylinder,sphere,and cubes.

The non-dimensional factor is called Fourier number F0. it signifies the degree of penitration of heating and cooling effect through a solid. using non dimensional terms ;

Measurement of temperature by a thermocouple is an important application of the lumped parameter analysis. The response of a thermocouple is defined as the time required for the thermocouple to attain the source temperature. It is the avident from eqn. that larger the quantity the faster the exponential term will be approach zero or the more rapid will be the response of the Temperature measuring device. This can be accomplished either by increasing the value of “h”or by decresing the wire diameter,density,and the specific heat.

Hence thin wire is recommended for use in thermocouple to ensure a rapid response. The quantity is called time constant is denoted by the symbol and

Thus is the time required for the temperature change to reach 36.8percent of its final value in response to a step change in temperature. In other words temperature difference would be reduced by 63.2%. The time required by a thermocouple to reach its 63.2%of the value of intital temperature difference is called its sensitivity. Depending upon the type of fluid,the response times different sizes of thermocouple wires usually vary between 0.04 to 2.5seconds.

As shown in the fig. consider the heating and cooling of a plane wall having a thickness of 2L and extending to infinitely in Y and Z direction. Let us assume that the wall,initially is at uniform temperature Ti and both the surfaces are suddenly exposed to and maintained at the ambient temperature Ta,the differential equation is The boundary condition are ; 1. A t =0 2. At x=0

At The solution obtained after rigorous mathematical analysis indicate that Form the eqn. it is evident that when conduction resistance is not negligible,the temperature history becomes a function of biot number Fourier number and the dimensionless parameter (x/l) Which indicates the location of point within the the plate where temperature is to be obtained.

The dimensionless parameter (x/l) is replaced by (r/R) in case of cylinder and spheres. From the eq, graphical charts have been prepared in a variety of forms.in the fig.is shown the heisler charts which depict the dimensionless temperature versus F0 for various values of (1/Bi) for solids of different geomatrical shapes such plates,cylinders,and the spheres. This charts provide the temperature history of the solids at its mid-planes(x=0);temperature at other locations are worked out by multiplying the mid plane temperature by correction factors and read from charts given in figures.

The following relationship is used: The values Bi and F0 as used in heisler charts are evaluted 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 the spheres. When both conduction and convection resistance are almost of equal importance the heisler charts extensively used to determine the temperature distribution.