 मैं सेल प्रुफिसर एसन खुलकरनें, असिष्टन प्रुफिसर इन दिपार्मेंद अप मैकनिकल इंजनिंग, वाल्चन इस्टिटो अप टेकनोजी शोलापूर, तोड़ा आम गुं तु कंड़क्त वीडिवो सेशन आं श्टरी श्टेट हीट कंड़क्शन तू. प्रुफिसर अनर्बूर the heat generated in time dow t equal to q generated equal to q dot into dx dy dz where q dot is the heat generated per unit volume and volume of the element is dx dy dz now this heat generated and stored within the element causes increase in internal energy of the solid element in time dow 2 so this increase in energy we can write as d is equal to rho into c into dx dy dz that is density into volume we can consider it as a mass so mass into specific we are expressing in terms of rho c into dx dy dz into dow t by dow 2 now combining the terms 1 to 4 as per the energy balance equation we get the equation as dow by dow x of kx into dow t by dow x plus dow by dow y of ky into dow t by dow y plus dow by dow z of kz into dow t by dow z plus q dot equal to rho c into dow t by dow 2 this equation is called as general heat conduction equation for anisotropic material means the material which have the thermal conductivity different in different direction whereas here we are consider kx ky and kz are different conductivities for x y and z direction for isotropic material means having a uniform same thermal conductivity as k is equal to kx equal to ky equal to kz this case is a practical case for homogeneous material the above equation gets simplified as taking k common and considering it as a constant k into dow t by dow x square plus dow 2 t by dow y square plus dow 2 t by dow z square plus q dot equal to rho into c into dow t by dow dow further this equation we can simplify as dividing by k throughout we get this equation as dow 2 t by dow x square plus dow 2 t by dow y square plus dow 2 t by dow z square plus q dot upon k equal to rho c upon k into dow t upon dow dow now this ratio of rho c upon k we can replace in the form of one property and the property is called as a thermal diffusivity that is equal to alpha is equal to ratio of k upon rho c which is the ratio of thermal conductivity upon heat capacity and if we replace dow 2 t by dow x square plus dow 2 by dow y square plus dow 2 by dow z square using a Laplace operator del square the general heat conduction equation in Cartesian coordinates we can write del square t plus q dot upon k equal to 1 upon alpha into dow t upon dow dow this is a simplified form of general heat conduction equation for isotropic material now if we consider the different cases case one if we consider the heat conduction without heat generation then the above equation reduces without heat generation means q dot will be equal to 0 this of the above equation becomes del square t equal to 1 upon alpha into dow t upon dow dow this equation is called as Fourier equation the case number two if we consider as the steady state heat conduction with heat generation now when we say steady state the change in temperature with respect to time will be equal to 0 and the above equation reduces to del square t plus q dot by k equal to 0 this equation is called as Poisson's equation and case number three we can write it as case number three as the conduction steady state conduction without heat generation now in this case we can say that whenever the steady state means temperature change with time is 0 dow t by dow dow 0 and without heat generation means q dot equal to 0 so the above equation reduces to del square t equal to 0 and this equation we call it as Laplace equation so these three equations we have obtained from general heat conduction equation now we will see the property called as thermal diffusivity and its significance now we have already defined the property called as thermal diffusivity and we have defined it as alpha equal to k upon root c if we prove the proper units the unit of alpha comes out to be meter square per second putting the unit of k as watt per meter Kelvin heat of the material as joule per kg Kelvin we get now the significance of alpha is that we see that alpha is ratio of two properties thermal conductivity and heat storage capacity when we say that higher the value of alpha the more will be the heat transmitted through the body this may happen due to more thermal conductivity of the material or less heat storage capacity of the material and this will happen when the material is having lower heat storage capacity or higher thermal conductivity when the value of alpha is low it means that the more amount of heat will be absorbed within the body unless amount of heat will be further transmitted or conducted through the body the physical significance of the alpha can be understood that the steady state equation heat conduction equation does not contain alpha when it is being steady state the temperature distribution is governed by only one property and that is called as thermal conductivity k but whenever the system is not steady when it is unsteady state the combination of two properties that is thermal conductivity and rho c these two properties govern the temperature distribution distribution equation and which we have combinedly defined as thermal diffusivity references heat transfer by P.K. Nag Tata McGrawil Company and Fundamentals of heat and mass transfer by R.C. Sachadeva Thank you