Thermal Engineering 4th Module

PART A

Derivation of general three dimensional conduction equations in Cartesian coordinate

Generally the heat conduction problem consists of finding the temperature at any time and at

any point within a specified solid that has been heated to a known initial temperature distribution

and whose surface has been subjected to a known set of boundary conditions.

Consider a solid as shown in Fig 2-4 with heat conducting in and out of a unit volume in all

three coordinate directions x, y and z

Making energy balance (1)

Substituting all the values in equations [1] above general three dimensional heat conduction equation becomes

In the above equation the quantity (alpha) is known as thermal diffusivity of the material.

Lower value of thermal heat capacity means the energy moving through the material. would be absorbed to a lesser degree and used to raise the temperature of the material. This means more energy is available for further transfer

Discussion on 3-D conduction in cylindrical and spherical coordinate systems (No derivation).

Cylindrical Coordinates

Spherical Coordinates

Three dimensional heat conduction equation in spherical coordinates is given by,

SPECIAL FORMS OF HEAT CONDUCTION EQUATION

From equation (3) of section ■ 2.3 some special cases of particular interest are as follows.

1. Laplace equation Considering the three dimensional heat conduction equation in Cartesian Co-ordinates, we have

2. Poisson's equation In many cases, the temperature at any point in a material doesn' t change with time,

3. Fourier equation

For unsteady state heat transfer with no internal heat generation then equation (2) above reduces to

One-dimensional conduction equations in rectangular, cylindrical and spherical coordinates for plane and composite walls.

RECTANGULAR OR CARTESIAN CO-ORDINATES

Consider a one dimensional system as shown in Fig 2-1. In the steady state system, the temperature doesn't change with time. If the temperature changes with time the system is known as unsteady state system. This is the general case where the temperature is not constant

(qgen) (q x+dex)

The above equation is known as one dimensional heat conduction equation.

CYLINDRICAL CO-Ordinate’s

The Cartesian coordinate system discussed above is not applicable to determine heat conduction in cylinders, cones, spheres etc. When heat conduction takes place through such geometries, cylindrical co-ordinate systems are used, since co-ordinate surfaces coincide with the boundary surfaces of the region. . For heat transfer analysis, consider an infinitesimal cylindrical volume element shown in figure 2.2.

The following assumptions are made while deriving the heat conduction equation

·       Thermal conductivity k, density

and specific heat C for the material do not change with position

·       Heat generation rate is uniform per unit volume per unit time

SPHERICAL CO-ORDINATES

Consider an infinitesimal spherical element of volume dV shown in fig. Considering heat conduction only along the direction r, we can derive heat conduction equation in a single co-ordinate

General equation for one dimensionil heat conduction T

The one dimensional heat conduction equation in the Cartesian (rectangular), cylindrical, and spherical coordinate systems is given by a single general equation as

Overall heat transfer coefficient.

In many instances it is customary to express the heat flow rate in the cases of single or multi-layered plane walls and cylinders with convection at the boundaries in terms of an overall conductance or overall heat transfer coefficient U.

A. PLANE WALL Consider a plane wall exposed to a hot fluid A on one side and a cold fluid B on the other side. The heat transfer is expressed as

The overall heat transfer coefficient due to combined heat transfer by convection and conduction is given as,

B. HOLLOW CYLINDER

Consider a hollow cylindrical tube with a hot fluid A flowing inside it and a cold fluid B flowing outside its surface. Let T_a, T_b be the corresponding temperatures and h_u, h_b be the corresponding heat transfer coefficients. The arrangement with an equivalent electric circuit is shown in Fig. 3-13

The heat flow rate is given by,

where V_a and V_b are the inside and outside overall heat transfer coefficients based on the respective inside and outside areas of the cylinder or tube.

Thermal contact resistance

Consider two solid bars brought into contact as shown in Fig. 3-10. The sides of the bars are insulated so that heat flows only in axial direction. The temperature profile through the solids experiences a sudden drop across the interface between the two materials. This temperature drop at the contact plane between the two materials is due to thermal contact resistance.

Consider the enlarged view of the interface as shown in Fig. 3-10.The direct contact between the

solids takes place only between a few spots whereas the gap between the solids is either filled with air or surrounding fluid. Since radiation effects are negligible at normal temperature and since there can not be any convection in such a thin layer of the fluid, heat transfer through the fluids filling the gaps or voids takes place mainly by conduction. Thus two principal contribution to the heat transfer at the contact surface are

1. The solid to solid conduction at the point of contact

2. The conduction through fluids filling the gaps or voids created by contact

Part B

Free or Natural Convection: Application of dimensional analysis for free convection-

When the heat transfer takes place by actual motion of the molecules without external assistance then heat transfer by convection is known as free convection

physical significance or Grashoff number;

use of correlations of free convection in vertical, horizontal and inclined flat plates, vertical and horizontal cylinders and spheres,

two more equations are proposed by Churchill1 and other for laminar flow for all values of Prandtl Number

HORIZONTAL PLATE

The average Nusselt number for free convection on a horizontal plate depends on whether the plate surface is warmer or cooler than the surrounding fluid and whether the surface is facing up or down.

VERTICAL CYLiNDER, If the thickness of the thermal boundary layer is much smaller than the cylinder radius, then the average Nusselt number for free convection on a vertical cylinder is same as that of a vertical plate.