Chapter 2 Heat Conduction Equation and Boundary Conditions
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Course
MECH369
Institution
MECH369
Chapter 2 presents a comprehensive exploration of the heat conduction equation and its boundary conditions, which form the foundation of understanding heat transfer through solid materials. The document delves into the underlying principles, mathematical formulations, and practical implications of ...
chapter 2 heat conduction equation and boundary co
one dimensional heat conduction
multi dimensional heat conduction
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MECH369
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Forsberg Heat Transfer
Chapter 2
Heat Conduction Equation
and Boundary Conditions
08-Feb-23
Heat Conduction Equation
Rectangular Coordinates
Infinitesimal Element
with heat flows
08-Feb-23
1
, Energy Balance on Element
Net rate of heat in = Rate of increase in internal energy
T
(qx + qy + qz ) − (qx +dx + qy +dy + qz +dz ) + qgen dx dy dz = (dx dy dz)c
t
where: qgen = volumetric rate of heat generation
= density of material
c = specific heat of material
T = temperature
t = time
08-Feb-23
Heat Conduction Equation (Cont’d)
T
For the x direction, qx = −k Ax and Ax = dy dz
x
qx +dx = qx + ( qx ) dx Similarly for y and z directions.
x
T
(qx + qy + qz ) − (qx +dx + qy +dy + qz +dz ) + qgen dx dy dz = (dx dy dz)c
t
becomes:
Heat Conduction Equation, Rectangular Coordinates
T T T T
k + k + k + qgen = c
x x y y z z t
08-Feb-23
2
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