Upload
anonymous-dwtqwj4qj
View
216
Download
0
Embed Size (px)
Citation preview
8/17/2019 Heat Trans Ch 4-2
1/32
University of Hail
Faculty of Engineering
DEPARTMENT OF MECHANICAL ENGINEERING
ME 315 – Heat TransferLecture notes
Chapter 4
Part 2Numerical analysis of conduction
Prepared by : Dr. N. Ait Messaoudene
Based on:
“Introduction to Heat Transfer”
Incropera, DeWitt, Bergman, and Lavine,
5th Edition, John Willey and Sons, 2007.
2
nd
semester 2011-2012
http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204http://ssb.kfupm.edu.sa/prod/bwckctlg.p_disp_course_detail?cat_term_in=200910&subj_code_in=ME&crse_numb_in=204
8/17/2019 Heat Trans Ch 4-2
2/32
The Finite-Difference Method
• An approximate method for determining temperatures at discrete
(nodal ) points of the physical system.
• Procedure:
– Represent the physical system by a nodal network (discrete number of points
representing the center of regular regions). Each node represents a certain region,
and its temperature is a measure of the average temperature of the region.
•The selection of nodal points is rarely arbitrary, depending often on matters
such as geometric convenience and the desired accuracy.
•The numerical accuracy of the calculations depends strongly on the number of
designated nodal points. If this number is large (a fine mesh), accurate solutions
can be obtained.
– Use the energy balance method to obtain a finite-difference equation for each
node of unknown temperature.
– Solve the resulting set of algebraic equations for the unknown nodal temperatures.
•Special treatment for points on the boundaries where conditions are prescribed
8/17/2019 Heat Trans Ch 4-2
3/32
The Nodal Network
The nodal network identifies discrete
points at which the temperature is to be
determined and uses an m,n notation todesignate their location.
The aggregate of points is termed a nodal
network, grid, or mesh.
A finite-difference approximation
is used to represent temperature
gradients in the domain.
8/17/2019 Heat Trans Ch 4-2
4/32
Finite-Difference Form of the Heat Equation
the value of the second derivative in the x direction at the (m, n) nodal point may be
approximated as
Use the
expressions
in the figure
Similarly:
If a regular grid is used (Δ x = Δy ), substitution of the above expressions in the 2D
steady state heat equation (with no heat generation) yields:
Or simply that the temperature of an interior node be equal to the average of the
temperatures of the four neighboring nodes.
8/17/2019 Heat Trans Ch 4-2
5/32
The Energy Balance Method
• As the direction of heat flow is often unknown, assume all heat flows are into the
nodal region of interest, and express all heat rates accordingly. Hence, the energy
balance (with heat generation) becomes:
(4.30)
• Consider application to an interior nodal point (one that exchanges heat by
conduction with four, equidistant nodal points):
where, for example,
(4.31)
Per unit length in the z direction
The assumption of heat flow into (m, n) requires assuming T m.n less then all 4 adjoining
temperatures so that it is substracted from these temperatures in the 4 expressions of heat ratesto (m, n).
In fact, the final values of temperatures will insure the correct direction of heat flow since it is
impossible for all 4 to be into (m, n) in steady state (unless we have a heat sink).
The remaining conduction
rates may be expressed as
8/17/2019 Heat Trans Ch 4-2
6/32
Substituting the heat rates into the energy balance and if Δ x = Δ y , it follows that the finite-
difference equation for an interior node (for a regular square grid) with generation is:
4.35
Which is the same as Equ. 4.29 when there is no generation.
So what is the interest or the advantage of this method?
In fact, this method is most powerful in more difficult cases as in:
•Boundaries of irregular shapes and mixed conditions,
•Problems involving multiple materials,
•embedded heat sources.
8/17/2019 Heat Trans Ch 4-2
7/32
To illustrate this, consider the node corresponding to an internal corner that exchanges
energy by convection with an adjoining fluid at T ∞
.
The 4 conduction heat rates may be expressed as
The total convection rate (also supposed into (m, n)may be expressed as:
Implicit in this expression is the assumption that the
exposed surfaces of the corner are at a uniformtemperature corresponding to the nodal temperature T m,n.
In the case of no generation, steady-state and regular square grid, the heat balance yields
(sum of all rates=0):
8/17/2019 Heat Trans Ch 4-2
8/32
• A summary of finite-difference equations for common nodal regions is provided
in Table 4.2.
Consider an external
corner with convectionheat transfer (Case 4).
1, , , 1 , , 0m n m n m n m n m nq q q
1, , , 1 ,
, ,
2 2
x yh h 02 2
m n m n m n m n
m n m n
T T T T y xk k
x y
T T T T
or, with , x y
1, , 1 ,
h h2 2 1 0m n m n m n
x xT T T T
k k
(4.43)
8/17/2019 Heat Trans Ch 4-2
9/32
where
8/17/2019 Heat Trans Ch 4-2
10/32
8/17/2019 Heat Trans Ch 4-2
11/32
If the energy balance is
satisfied, the left-hand
side of this equation will
be identically
equal to zero. Substituting
values, we obtain The inability to precisely satisfy
the energy balance can be
attributable to temperature
measurement errors, the
approximations employed in
developing the finite-difference
equations, and the use of arelatively coarse mesh.
8/17/2019 Heat Trans Ch 4-2
12/32
• Note potential utility of using thermal resistance concepts to express rate
equations. E.g., conduction between adjoining dissimilar materials with
an interfacial contact resistance.
(4.46)
thermal contact resistance(in m2.K/W)
8/17/2019 Heat Trans Ch 4-2
13/32
Solutions Methods
• Matrix Inversion: Expression of system of N finite-difference equations for
N unknown nodal temperatures as:
A T C (4.48)
Coefficient
Matrix (NxN)Solution Vector
(T1,T2, …TN)Right-hand Side Vector of Constants
(C1,C2…CN)
Solution 1
T A C
Inverse of Coefficient Matrix (4.49)
• Gauss-Seidel Iteration: Each finite-difference equation is written in explicit
form, such that its unknown nodal temperature appears alone on the left-
hand side: 1 ( 1)
1 1
i N ij ijk k k ii j j
j j iii ii ii
a aC T T T
a a a
(4.51)
where i =1, 2,…, N and k is the level of iteration.
Iteration proceeds starting from initial assumed values for each temperature
T i until a prescribed convergence criterion is satisfied for all nodes:
1k k i iT T
Acceptable error in
temperature values
8/17/2019 Heat Trans Ch 4-2
14/32
Verifying the Accuracy of the Solution
It is good practice to verify that a numerical solution has been correctly formulated by
performing an energy balance on a control surface surrounding all nodal regions
whose temperatures have been evaluated.
The temperatures should be substituted into the energy balance equation, and if the
balance is not satisfied to a high degree of precision, the finite difference equationsshould be checked for errors. If no errors are found, the grid should be refined.
8/17/2019 Heat Trans Ch 4-2
15/32
8/17/2019 Heat Trans Ch 4-2
16/32
04500:5 Node 5763 T T T T
8/17/2019 Heat Trans Ch 4-2
17/32
8/17/2019 Heat Trans Ch 4-2
18/32
EXAMPLE 4 4
8/17/2019 Heat Trans Ch 4-2
19/32
EXAMPLE 4.4
A major objective in advancing gas turbine engine technologies is to increase the temperature limit
associated with operation of the gas turbine blades. This limit determines the permissible turbine gas inlet
temperature, which, in turn, strongly influences overall system performance. In addition to fabricating
turbine blades from special, high-temperature, high-strength superalloys, it is common to use internal
cooling by machining flow channels within the blades and routing air through the channels. We wish to
assess the effect of such a scheme by approximating the blade as a rectangular solid in which rectangular
channels are machined. The blade, which has a thermal conductivity of k = 25 W/m.K, is 6 mm thick, and
each channel has a 2mm x 6 mm rectangular cross section, with a 4-mm spacing between adjoining
channels.
8/17/2019 Heat Trans Ch 4-2
20/32
8/17/2019 Heat Trans Ch 4-2
21/32
8/17/2019 Heat Trans Ch 4-2
22/32
8/17/2019 Heat Trans Ch 4-2
23/32
8/17/2019 Heat Trans Ch 4-2
24/32
8/17/2019 Heat Trans Ch 4-2
25/32
8/17/2019 Heat Trans Ch 4-2
26/32
Problem 4.39: Finite-difference equations for (a) nodal point on a diagonal
surface and (b) tip of a cutting tool.
(a) Diagonal surface (b) Cutting tool.
Schematic:
ASSUMPTIONS: (1) Steady-state, 2-D conduction, (2) Constant properties
8/17/2019 Heat Trans Ch 4-2
27/32
ANALYSIS: (a) The control volume about node m,n is triangular with sides x and y and diagonal
(surface) of length 2 x.
The heat rates associated with the control volume are due to conduction, q1 and q2, and to convection,
qc. An energy balance for a unit depth normal to the page yields
inE 0
1 2 cq q q 0
m,n-1 m,n m+1,n m,n m,nT T T T
k x 1 k y 1 h 2 x 1 T T 0.y x
With x = y, it follows that
m,n-1 m+1,n m,n
h x h xT T 2 T 2 2 T 0.
k k
(b) The control volume about node m,n is triangular with sides x/2 and y/2 and a lower diagonal
surface of length 2 x/2 .
The heat rates associated with the control volume are due to the uniform heat flux, qa, conduction, q b,
and convection qc. An energy balance for a unit depth yields
ina b c
E =0q q q 0
m+1,n m,no m,nT Tx y x
q 1 k 1 h 2 T T 0.2 2 x 2
or, with x = y,
m+1,n o m,nh x x h x
T 2 T q 1 2 T 0.
k k k
8/17/2019 Heat Trans Ch 4-2
28/32
Problem 4.76: Analysis of cold plate used to thermally control IBM multi-chip,
thermal conduction module.
Features:
• Heat dissipated in the chips is transferredby conduction through spring-loaded
aluminum pistons to an aluminum cold
plate.
• Nominal operating conditions may be
assumed to provide a uniformlydistributed heat flux of
at the base of the cold plate.
5 210 W/mo
q
• Heat is transferred from the cold
plate by water flowing through
channels in the cold plate.
Find: (a) Cold plate temperature distribution
for the prescribed conditions. (b) Options
for operating at larger power levels while
remaining within a maximum cold plate
temperature of 40C.
8/17/2019 Heat Trans Ch 4-2
29/32
Schematic:
ASSUMPTIONS: (1) Steady-state conditions, (2) Two-dimensional conduction, (3) Constant properties.
8/17/2019 Heat Trans Ch 4-2
30/32
ANALYSIS: Finite-difference equations must be obtained for each of the 28 nodes. Applying the energy
balance method to regions 1 and 5, which are similar, it follows that
ode 1: 2 6 1 0 y x T x y T y x x y T
ode 5: 4 10 5 0 y x T x y T y x x y T
Nodal regions 2, 3 and 4 are similar, and the energy balance method yields a finite-difference equation of
the form
odes 2,3,4:
1, 1, , 1 ,2 2 0m n m n m n m n y x T T x y T y x x y T
Energy balances applied to the remaining combinations of similar nodes yield the following
finite-difference equations.
Nodes 6, 14: 1 7 6 x y T y x T x y y x h x k T h x k T 19 15 14 x y T y x T x y y x h x k T h x k T
Nodes 7, 15: 6 8 2 72 2 2 y x T T x y T y x x y h x k T h x k T
14 16 20 152 2 2 y x T T x y T y x x y h x k T h x k T
8/17/2019 Heat Trans Ch 4-2
31/32
Nodes 8, 16: 7 9 11 3y x T 2 y x T x y T 2 x y T 3 y x 3 x y
8h k x y T h k x y T
15 17 11 212 2 3 3 y x T y x T x y T x y T y x x y
16h k x y T h k x y T
Node 11: 8 16 12 112 2 2 x y T x y T y x T x y y x h y k T h y k T
Nodes 9, 12, 17, 20, 21, 22:
m 1,n m 1,n m,n 1 m,n 1 m,ny x T y x T x y T x y T 2 x y y x T 0
odes 10, 13, 18, 23:
1, 1, 1, ,2 2 0n m n m m n m n x y T x y T y x T x y y x T
ode 19: 14 24 20 192 2 0 x y T x y T y x T x y y x T
odes 24, 28: 19 25 24 o x y T y x T x y y x T q x k
23 27 28 o x y T y x T x y y x T q x k
Nodes 25, 26, 27:
1, 1, , 1 ,2 2 2m n m n m n m n o y x T y x T x y T x y y x T q x k
8/17/2019 Heat Trans Ch 4-2
32/32
Evaluating the coefficients and solving the equations simultaneously, the steady-state temperature
distribution (C), tabulated according to the node locations, is:
23.77 23.91 24.27 24.61 24.74
23.41 23.62 24.31 24.89 25.07
25.70 26.18 26.3328.90 28.76 28.26 28.32 28.35
30.72 30.67 30.57 30.53 30.52
32.77 32.74 32.69 32.66 32.65
(b) For the prescribed conditions, the maximum allowable temperature (T24 = 40C) is reached when
oq = 1.407 105 W/m2 (14.07 W/cm2).
Options for extending this limit could include use of a copper cold plate (k 400 W/mK) and/orincreasing the convection coefficient associated with the coolant.
With k = 400 W/mK, a value of oq = 17.37 W/cm2 may be maintained.
. With k = 400 W/mK and h = 10,000 W/m2K (a practical upper limit), oq = 28.65 W/cm2.
Additional, albeit small, improvements may be realized by relocating the coolant channels closer to the
base of the cold plate.