This preview shows page 1. Sign up to view the full content.
PROBLEM 4.39
KNOWN:
Nodal point configurations corresponding to a diagonal surface boundary subjected to a
convection process and to the tip of a machine tool subjected to constant heat flux and convection
ooling.
c
F
IND:
Finitedifference equations for the node m,n in the two situations shown.
SCHEMATIC:
A
SSUMPTIONS:
(1) Steadystate, 2D conduction, (2) Constant properties.
ANALYSIS:
(a) The control volume about node m,n has triangular shape with sides
Δ
x and
Δ
y while
the diagonal (surface) length is
2
Δ
x.
The heat rates associated with the control volume are due to
onduction, q
1
and q
2
, and to convection, q
c
.
Performing an energy balance, find
c
()
in
out
1
2
c
m,n1
m,n
m+1,n
m,n
m,n
E
E
0
q
q
q
0
TT
T
T
kx
1
ky
1
h2
x
1T T
0
yx
∞
−=
+
+
=
−−
Δ⋅
+ Δ⋅
+
−
=
ΔΔ
±±
.
Note that we have considered the solid to have unit depth normal to the page.
Recognizing that
Δ
x =
y, dividing each term by k and regrouping, find
Δ
m,n1
m+1,n
m,n
hx
T
T
T
2
T
0.
kk
∞
⎡⎤
++
⋅
−
+
⋅
⎢⎥
⎣⎦
=
<
(b) The control volume about node m,n has triangular shape with sides
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/07/2010 for the course MAE Heat Trans taught by Professor Lee,j.s. during the Spring '10 term at Seoul National.
 Spring '10
 LEE,J.S.
 Heat Transfer

Click to edit the document details