Heat Trans Ch 4-2

8/17/2019 Heat Trans Ch 4-2

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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


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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


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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.


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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.


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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


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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.


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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):


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• 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)


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where


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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.


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• 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)


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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


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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.


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04500:5 Node 5763 T T T T


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EXAMPLE 4 4


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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.


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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


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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


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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.


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Schematic:

ASSUMPTIONS: (1) Steady-state conditions, (2) Two-dimensional conduction, (3) Constant properties.


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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


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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


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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.

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Heat Trans Ch 4-2