handbook of computational analytical heat...

Handbook of Computational Analytical Heat Conduction X20B1T0 problem

Filippo de Monte, James V. Beck, et al. – January 27, 2013 1

X20B1T0 problem

Contents 1. Problem description …………………………………………………… 2 2. Dimensional governing equations ……………………………………. 2 3. Dimensionless variables ………………………………………………. 3 4. Dimensionless governing equations ………………………………….. 3 5. Dimensionless temperature and heat flux solutions ………………… 3 6. Plots and tables of dimensionless temperature and heat flux ………. 4 Appendix. Matlab function ……………………………………………….. 10

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Filippo de Monte, James V. Beck, et al. – January 27, 2013 1. Problem description Semi-infinite Cartesian body, initially at zero temperature and with temperature-independent properties, subject to a step change in heat flux at x = 0, as depicted in Fig. 1.

Fig. 1 – Schematic of the 1D transient X20B1T0 problem. 2. Dimensional governing equations

The mathematical formulation of the problem is

2

2

1T Tx tα

∂ ∂=∂ ∂

( 0 < x < ∞ ; 0t > ) (1a)

−k

∂T∂x

⎛⎝⎜

⎞⎠⎟ x=0

= q0 ( 0t > ) (1b)

T (∞,t) = finite ( 0t > ) (1c)

T (x,0) = 0 ( 0 < x < ∞ ) (1d)



3. Dimensionless variables We have a total of four dimensionless groups

T = T

q0L / k,

q = q

q0

= − ∂ T∂ x

⎛⎝⎜

⎞⎠⎟

, x = x

L,

t = αt

L2 = Fo , (2)

where q0L / k is the steady-state temperature of the X21B10T0 problem at x = 0,

x ∈[0,1] and Fo is the well-known Fourier number ( ≥ 0 ). 4. Dimensionless governing equations

The mathematical formulation in dimensionless form is

∂2 T∂ x2 = ∂ T

∂t ( 0 < x < ∞ ; t > 0 ) (3a)

− ∂ T

∂ x⎛⎝⎜

⎞⎠⎟ x=0

= 1 ( t > 0 ) (3b)

T (∞, t ) = finite ( t > 0 ) (3c)

T ( x,0) = 0 ( 0 < x < ∞ ) (3d)

5. Dimensionless temperature and heat flux solutions

The solution to the current X20B1T0 problem is a well-established exact analytical solution available in the heat conduction literature [1]. The solution is unique and is given in Ref. [1, p. 75, Eq. (6)] as

T ( x, t ) ≈ 2 t ierfc

x

2 t⎛⎝⎜

⎞⎠⎟

( 0 ≤ x ≤ ∞ )

(4a)

q( x, t ) ≈ erfc

x

2 t⎛⎝⎜

⎞⎠⎟

( 0 ≤ x ≤ ∞ )

(4b)

where ierfc(z) is the so-called complementary error function integral defined as

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Filippo de Monte, James V. Beck, et al. – January 27, 2013

2

ierfc( ) erfc( )zez z zπ

−

= − (4c)

As the above function is not available in Matlab ambient, a simple code is given in Appendix for computing it.

6. Plots and tables of dimensionless temperature and heat flux

By using the exact analytical solutions listed before and implemented in Matlab ambient (see Appendix), 2D plots, 3D plots and tables of temperature and heat flux can be derived. In detail,

2D plots of temperature and heat flux are shown in Figs. 2 and 3.

3D plots of temperature and heat flux are given in Figs. 4 and 5.

Tables 1 and 2 provide numerical values of temperature and heat flux.



Fig. 2 – Temperature graphs.

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Fig. 3 – Heat flux graphs.



Fig. 4 – Temperature surface plot.

Fig. 5 – Heat flux surface plot.

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Filippo de Monte, James V. Beck, et al. – January 27, 2013 Table 1 – Numerical values of temperature.

t x = 0 x = 0.25 x = 0.50 x = 0.75 x = 1.0 0.000 0.001 0.005 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 ∞

0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.03568248 0.00000000 0.00000000 0.00000000 0.00000000 0.07978846 0.00040083 0.00000001 0.00000000 0.00000000 0.11283792 0.00437714 0.00001435 0.00000000 0.00000000 0.15957691 0.02023475 0.00080165 0.00000841 0.00000002 0.19544100 0.03923836 0.00372182 0.00015021 0.00000242 0.22567583 0.05851013 0.00875429 0.00070171 0.00002870 0.25231325 0.07729749 0.01536581 0.00187304 0.00013467 0.27639532 0.09540466 0.02307320 0.00373774 0.00039278 0.29854107 0.11280699 0.03152322 0.00627834 0.00086736 0.31915382 0.12953601 0.04046949 0.00943448 0.00160331 0.33851375 0.14564025 0.04974103 0.01313143 0.00262548 0.35682482 0.16117091 0.05921833 0.01729430 0.00394265 0.37424103 0.17617663 0.06881697 0.02185408 0.00555210 0.39088201 0.19070170 0.07847673 0.02674978 0.00744363 0.40684289 0.20478578 0.08815426 0.03192860 0.00960255 0.42220082 0.21846413 0.09781819 0.03734537 0.01201175 0.43701937 0.23176806 0.10744571 0.04296160 0.01465314 0.45135167 0.24472535 0.11702026 0.04874464 0.01750858 0.46524265 0.25736072 0.12652984 0.05466671 0.02056042 0.47873074 0.26969628 0.13596587 0.06070424 0.02379186 0.49184908 0.28175179 0.14532230 0.06683715 0.02718715 0.50462650 0.29354503 0.15459499 0.07304830 0.03073162 0.56418958 0.34908866 0.19964123 0.10483226 0.05025454 0.61803872 0.39995170 0.24250559 0.13706754 0.07189292 0.66755812 0.44714004 0.28333988 0.16915845 0.09479979 0.71364965 0.49134650 0.32234183 0.20081351 0.11843665 0.75693976 0.53307134 0.35970266 0.23189248 0.14245623 0.79788456 0.57268940 0.39559311 0.26233384 0.16663094 0.83682839 0.61048999 0.43016161 0.29211828 0.19080971 0.87403874 0.64670189 0.46353612 0.32124946 0.21489142 0.90972837 0.68150964 0.49582693 0.34974346 0.23880825 0.94406974 0.71506468 0.52712939 0.37762276 0.26251495 0.97720502 0.74749307 0.55752632 0.40491289 0.28598182 1.00925301 0.77890106 0.58709006 0.43164041 0.30918997 1.04031419 0.80937924 0.61588406 0.45783180 0.33212817 1.07047447 0.83900551 0.64396438 0.48351274 0.35479054 1.09980797 0.86784749 0.67138070 0.50870787 0.37717507 1.12837917 0.89596432 0.69817732 0.53344049 0.39928246

∞ ∞ ∞ ∞ ∞



Table 2 – Numerical values of heat flux.

t x = 0 x = 0.25 x = 0.50 x = 0.75 x = 1.0 0.000 0.001 0.005 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 ∞

0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 1.00000000 0.00000002 0.00000000 0.00000000 0.00000000 1.00000000 0.01241933 0.00000057 0.00000000 0.00000000 1.00000000 0.07709987 0.00040695 0.00000011 0.00000000 1.00000000 0.21129955 0.01241933 0.00017683 0.00000057 1.00000000 0.30743417 0.04122683 0.00219965 0.00004456 1.00000000 0.37675912 0.07709987 0.00800994 0.00040695 1.00000000 0.42919530 0.11384630 0.01770607 0.00156540 1.00000000 0.47048642 0.14891467 0.03038282 0.00389242 1.00000000 0.50403587 0.18144921 0.04502088 0.00752632 1.00000000 0.53197106 0.21129955 0.06079272 0.01241933 1.00000000 0.55568979 0.23859283 0.07709987 0.01842213 1.00000000 0.57615012 0.26355248 0.09353251 0.02534732 1.00000000 0.59403234 0.28642202 0.10981941 0.03300626 1.00000000 0.60983404 0.30743417 0.12578642 0.04122683 1.00000000 0.62392846 0.32679957 0.14132600 0.04986020 1.00000000 0.63660163 0.34470422 0.15637572 0.05878172 1.00000000 0.64807687 0.36131043 0.17090352 0.06788915 1.00000000 0.65853137 0.37675912 0.18489760 0.07709987 1.00000000 0.66810774 0.39117252 0.19835953 0.08634782 1.00000000 0.67692224 0.40465676 0.21129955 0.09558070 1.00000000 0.68507066 0.41730417 0.22373335 0.10475749 1.00000000 0.69263278 0.42919530 0.23567991 0.11384630 1.00000000 0.72367361 0.47950012 0.28884437 0.15729921 1.00000000 0.74688563 0.51860502 0.33292161 0.19670560 1.00000000 0.76508719 0.55009732 0.37002771 0.23199772 1.00000000 0.77985462 0.57615012 0.40173564 0.26355248 1.00000000 0.79214739 0.59816145 0.42919530 0.29184055 1.00000000 0.80258735 0.61707508 0.45325470 0.31731051 1.00000000 0.81159751 0.63355348 0.47454914 0.34035574 1.00000000 0.81947698 0.64807687 0.49356279 0.36131043 1.00000000 0.82644401 0.66100285 0.51067082 0.38045513 1.00000000 0.83266211 0.67260382 0.52616848 0.39802472 1.00000000 0.83825649 0.68309140 0.54029137 0.41421618 1.00000000 0.84332489 0.69263278 0.55322997 0.42919530 1.00000000 0.84794489 0.70136205 0.56513999 0.44310233 1.00000000 0.85217893 0.70938812 0.57615012 0.45605654 1.00000000 0.85607790 0.71680052 0.58636771 0.46815991 1.00000000 0.85968380 0.72367361 0.59588309 0.47950012

1.00000000 1.00000000 1.00000000 1.00000000 1.00000000

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Filippo de Monte, James V. Beck, et al. – January 27, 2013 Appendix. Matlab function

fdX20B1T0

Heat conduction function for the X20B1T0 case.

Syntax

[Td,qd] = fdX20B1T0(xd,td)

Description

fdX20B1T0 (xd, td) returns the dimensionless temperature Td and heat flux qd solutions at a given dimensionless location xd from the heated surface, between 0 and infinity, and at a given dimensionless time td, for the X20B1T0 problem. If xd and td are not single values but arrays (length(xd) = n and length(td) = m) defining the dimensionless locations and times of interest, respectively, the above function returns the dimensionless temperature Td and heat flux qd as double subscripted arrays, where size(Td) = size(qd) = [m, n].

Examples Example 1 >> [Td, qd]=fdX20B1T0(.25,.1) Td = 0.161170914709907 qd = 0.576150122030579



Example 2

>> xd=[0.1 0.5 0.7]' xd = 0.100000000000000 0.500000000000000 0.700000000000000 >> td=[0.01 0.2]' td = 0.010000000000000 0.200000000000000 >> [Td, qd]=fdX20B1T0(xd,td) Td = 0.039928245674849 0.000014352414313 0.000000019773817 0.410921227173961 0.154594987182527 0.085637493624527 qd = 0.479500122186953 0.000406952017445 0.000000743098372 0.874367061162892 0.429195300440349 0.268381627292761

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Matlab function: fdX20B1T0.m % fdX20B1T0 function % Revision History % 1 21 2013 James V. Beck, Matthew Lempke and Filippo de Monte % INPUTS: % xd: dimensionless location starting at xd=0 % td: dimensionless time starting at td=0 % OUPUTS: % Td: dimensionless temperature calculated at (xd,td) % qd: dimensionless heat flux calculated at (xd,td) % Calling Sequence: % ierfc(z) for computing the complementray error function integral function [Td,qd]=fdX20B1T0(xd,td) lengthx=length(xd); lengtht=length(td); Td=zeros(lengtht,lengthx); % Preallocating Arrays for speed qd=zeros(lengtht,lengthx); % Preallocating Arrays for speed for it=1:lengtht % Begin time loop td_it=td(it); % Set current time for ix=1:lengthx % Begin space loop xd_ix=xd(ix); % Set current space if td_it == 0 % For time t=0 condition Td(it,ix)=0; % Set inital temperature qd(it,ix)=0; % Set inital heat flux else % Solution at any time Td(it,ix)=sqrt(4*td_it)*ierfc(xd_ix/sqrt(4*td_it)); qd(it,ix)=erfc(xd_ix/sqrt(4*td_it)); end % if td_it end % for ix end % for it

function [ierfc]=ierfc(z) ierfc =(1/sqrt(pi))*exp(-z^2)-z*erfc(z);

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