Derive generalized heat conduction equation and state the boundary conditions used to obtain the solution.
Answers
The basic form of heat conduction equation is obtained by applying the first law of thermodynamics (principle of conservation of energy).
What is the meaning of each term in the equation? What is heat generation?
The first term in the above equation represents the rate of heat energy coming into the element at x, y and z planes. The second term represents the rate of heat energy coming out of the element at x+dx, y+dy and z+dz planes.The third term represents the rate of heat generation inside the element.
Now, what is heat generation?
A medium conducting heat energy may involve the conversion of mechanical, electrical, nuclear or chemical energy into heat energy. For example, when a resistance wire conducts electric current, it converts electrical energy into heat energy at a rate of I²R, where I is the current and R is the electrical resistance of the wire.
Similarly, heat is generated in an exothermic chemical reaction in a medium. The reaction may also be an endothermic reaction. In such a case, the heat generation term will become a negative quantity.
Likewise, nuclear fission process in a nuclear reactor generates a large amount of heat in fuel elements.
Heat is also generated in a nuclear fusion reaction. For example, hydrogen atoms get fused into helium making the sun a large nuclear reactor.
One thing to note is that heat generation is a volumetric phenomenon. This means that heat generation occurs throughout the body of the medium. For this reason, the rate of heat generation in a medium is usually specified per unit volume, W/m³.
The rate of heat generation may vary with respect to time as well as position within the medium. When variation with respect to position (x,y,z) is known, we can calculate total rate of heat generation in a medium of volume V by:
If the rate of heat generation is uniform throughout the medium then the above equation will become:
Now, comes the fourth term. It represents the rate of change of energy content of the mass contained in the element.
Derivation of heat conduction equation
In general, the heat conduction through a medium is multi-dimensional. That is, heat transfer by conduction happens in all three- x, y and z directions. In some cases, the heat conduction in one particular direction is much higher than that in other directions. In such cases, we approximate the heat transfer problems as being one-dimensional, neglecting heat conduction in other directions.
Now, we will develop the governing differential equation for heat conduction. Consider again the differential element of volume dV = dx*dy*dz in Cartesian coordinate system. Writing the mathematical form of the energy balance equation for the element,
Rate of heat generation within the element,
The rate of change of energy content of the element:
Or,
Now,
Similarly,
and,
Now, plug in the above terms in the energy balance equation and divide the equation by dx*dy*dz.
We will get:
This is the general heat conduction equation in Cartesian coordinates.
Special cases
(a) Steady state
Steady state refers to a stable condition that does not change over time. Time variation of temperature is zero. Hence,
(b) Uniform properties
If the material is homogeneous and isentropic, the thermal conductivity of the material would be constant.
{Comment: What do you mean by homogeneous and isentropic material? The term homogenous means, the values of physical properties of a material do not vary with position within the body of the material. E.g., The value of thermal conductivity at position (x1, y1, z1) will be same as that at some other position (x2, y2, z2). The term isentropic means, the value of physical properties at a point in different directions will be same. That is to say kx=ky=kz at a point. }
Or,
The above equation is also known as POISSON’S Equation.
(c ) No heat generation
When there is no heat generation inside the element, the differential heat conduction equation will become,
Or,
The above equation is also known as LAPLACE Equation.
(d) One dimensional form of equation
If heat conduction in any one direction is in dominance over heat conduction in other direct