SteadyState 2 D Conduction Chapter Four abridged 2

Steady-State 2 D Conduction Chapter Four (abridged)

2 D Steady-State Conduction • Graphical Solutions – Identify Isotherms – Identify Adiabats • Heat transfer in a long, prismatic solid with two isothermal surfaces and two insulated surfaces:

2 D Steady-State Conduction • Turning 2 D Geometries into pre-defined shapes – Use symmetry, and assume symmetry lines are adiabatic

2 D Steady-State Conduction • Numerical Solutions – Do not do these by hand – Can program equations in MATLAB or Python – Should probably use existing software to get approximations Ø Solidworks Ø EES Ø Other heat transfer software

Transient Conduction: The Lumped Capacitance Method Chapter Five Sections 5. 1 through 5. 3

Transient Conduction • A heat transfer process for which the temperature varies with time, as well as location within a solid. • It is initiated whenever a system experiences a change in operating conditions. • It can be induced by changes in: – surface convection conditions ( ), – surface radiation conditions ( ), – a surface temperature or heat flux, and/or – internal energy generation. • Solution Techniques – The Lumped Capacitance Method – Exact Solutions – The Finite-Difference Method

Examples Heat Treatment of Steel Quenching https: //www. youtube. com/watch? v=C 2 Hbk. Xlnojg https: //www. youtube. com/watch? v=di 3 f. D_z. Su. L 8 Annealing https: //www. youtube. com/watch? v=Zu. Zx. J 3 Ye. BWY

Biot Number The Biot Number and Validity of The Lumped Capacitance Method • The Biot Number: The first of many dimensionless parameters to be considered. Ø Definition: Ø Physical Interpretation: See Fig. 5. 4. Ø Criterion for Applicability of Lumped Capacitance Method:

Lumped Capacitance Method The Lumped Capacitance Method • Based on the assumption of a spatially uniform temperature distribution throughout the transient process. Hence, . • Why is the assumption never fully realized in practice? • General Lumped Capacitance Analysis: Ø Consider a general case, which includes convection, radiation and/or an applied heat flux at specified surfaces as well as internal energy generation

Lumped Capacitance Method (cont. ) Ø First Law: • Assuming energy outflow due to convection and radiation and inflow due to an applied heat flux (5. 15) • May h and hr be assumed to be constant throughout the transient process? • How must such an equation be solved?

Special Case (Negligible Radiation) • Special Cases (Exact Solutions, ) Ø Negligible Radiation The non-homogeneous differential equation is transformed into a homogeneous equation of the form: Integrating from t = 0 to any t and rearranging, (5. 25) To what does the foregoing equation reduce as steady state is approached? How else may the steady-state solution be obtained?

Special Case (Convection) Ø Negligible Radiation and Source Terms (5. 2) Note: (5. 6) The thermal time constant is defined as (5. 7) Thermal Lumped Thermal Resistance, Rt Capacitance, Ct The change in thermal energy storage due to the transient process is (5. 8)

Special Case (Radiation) Ø Negligible Convection and Source Terms Assuming radiation exchange with large surroundings and a = e, (5. 18) This result necessitates implicit evaluation of T(t).