Page 17 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

INTRODUCTION

m e x u 9

q x

∆x x
h P ∆x (T − T )
a
q x +dx
x + dx

m e x +dx

Figure 1.3 Conservation of energy in a moving body

the environment by radiation is assumed to be negligibly small. The energy is conducted,
convected and transported with the material in motion. With reference to Figure 1.3, we
can write the following equations of conservation of energy, that is,
q x + me x + GA x = q x+dx + me x+dx + hP x(T − T a ) (1.28)

where m is the mass flow, ρAu which is assumed to be constant; ρ, the density of the
material; A, the cross-sectional area; P , the perimeter of the control volume; G, the heat
generation per unit volume and u, the velocity at which the material is moving. Using a
Taylor series expansion, we obtain

de x dT
m(e x − e x+dx ) =−m x =−mc p x (1.29)
dx dx
Note that de x = c p dT at constant pressure. Similarly, using Fourier’s law
(Equation 1.1),
d dT
q x − q x+dx = kA (1.30)
dx dx
Substituting Equations 1.29 and 1.30 into Equation 1.28, we obtain the following con-
servation equation:
d dT dT
kA − hP(T − T a ) − ρc p Au + GA = 0 (1.31)
dx dx dx