Conduction Heat tranfer: Unteady tate
Chapter Objective For olving the ituation that Where temperature do not change with poition. In a imple lab geometry where temperature vary alo with poition. Near the urface of a large body (emi infinite region)
Keyword Internal reitance External reitance Biot number Lumped parameter anayi 1D and multi dimenional heat conduction Heiler chart Semi infinite region
1 Lumped Parameter Analyi r=r r=0 r=0.5r Surr Figure 1. Several temperature in the ytem. In tranient, r=r = r=0.5r = r=0?
Lumped Parameter Analyi Figure. A olid with convection over it urface. mc p Δ ha( )Δt (1)
Lumped Parameter Analyi mc p Δ ha( )ΔΔ (1) M : Ma C p : Specific heat h : Convective heat tranfer coefficient A : Surface area : Bulk fluid temperature ()
Lumped Parameter Analyi (3)
(4) (5)
Biot Number r=r r=0 r=0.5r Surr Figure 3. Several temperature in the ytem. So, when can we apply r=r = r=0.5r = r=0?
Biot Number Bi (Biot Number) : Deciding whether internal reitance can be ignored. (6) (7)
Characteritic Length Characteritic length V/A Path of leat thermal reitance Characteric length = emperature can be changed in hort time (8) Figure 4. Characteritic length for heat conduction in variou geometrie.
Example 1 What i the temperature of the egg after 60min? Figure 5. Schematic for Example 1. Known: Initial temperature of an egg Find: emperature of the egg after 60min. Given data: i = 0 air = 38 h = 5. W/m ㆍ K ρ = 1035 kg/m 3 C p = 3350 J/kgㆍK k = 0.6 W/mㆍK
Aumption: 1. Egg i approximately pherical.. Surface heat tranfer coefficient provided i an average value. 3. Lumped parameter analyi. Bi (Biot Number) = hv / Ak = 0.07 < 0.1 Being Bi <0.1, lumped analyi can be applied! Uing (Eqn. 5), hen, = 9.1
Aumption: 1. Egg i approximately pherical.. Surface heat tranfer coefficient provided i an average value. 3. Lumped parameter analyi. Bi (Biot Number) = hv / Ak = 0.07 < 0.1 Being Bi <0.1, lumped analyi can be applied! Uing (Eqn.5), hen, = 9.1
3 When Internal Reitance I Not Negligible r=r r=0 r=0.5r Surr Figure 1: Several temperature in the ytem. he ituation, r=r r=0.5r r=0 (i.e. Bi 0.1)
When Internal Reitance I Not Negligible Figure 6. Schematic of a lab howing the line of ymmetry at x = 0 and the two urfcae at x = L and at x = L maintained at temperature S. he material i very large (extend to infinity) in the other two direction.
When Internal Reitance I Not Negligible (9) Boundary condition (10) (11)
When Internal Reitance I Not Negligible Initial condition (1) (13) α (hermal diffuivity) = k/ρc p
How emperature Change with ime For viualizing emperature v. Poition and ime, infinite erie hould be implified Figure 7. he term in the erie (n = 0, 1,... in Equation 5.13) drop off rapidly for value of time. Calculation are for F O = 0.0048 at 30 and F O = 0.096 at 600 for a thickne of L = 0.03 m and a typical α = 1.44 x 10 7m/ for bio material.
How emperature Change with ime Comparing different term at each time(t= 30, t= 600), Contribution decay Gradually at t= 30 Rapidly at t= 600 (15) (16)
emperature Change with Poition and Spatial Average We can ee that temperature varie a a coine function herefore, we need to define patial average temperature (15) (16) t L i e L x co 4 t L L x i co 4 ln ln
Spatial average temperature av 1 L 0 L dx (17) Applying (5.17) to (5.16) give ln av i 8 ln L t (18)
emperature Change with Size t L 4 ln 8 av i (19)
Chart Developed from the Solution: heir Ue and Limitation. It can be een that temperature i a function of x/l and αt/l Chart are developed becaue of the complexity of the calculation of erie. (0) 1 0 1 co 1 1 4 L t n n n i e L x n n
Chart are developed with the condition of n=0. In other word, it i a plot of Eqn. 5 And it i alo called Heiler chart. here are ome aumption for the development of the chart. hee are: 1. Uniform initial temperature. Contant boundary fluid temperature 3. Perfect lab, cylinder or phere 4. Far from edge 5. No heat generation (Q=0) 6. Contant thermal propertie (k, α, c p are contant) 7. ypically for time long after initial time, given by αt/l >0.
Figure 8. Unteady tate diffuion in a large lab
Example. emperature Reached During Food Sterilization Figure 9. A cylindrical can containing food to be terilized. Surface temperature of a lab of tuna i uddenly increaed Find the temperature at the center of the lab after 30 min
Given data: 1. hickne of lab = 5 mm. hermal diffuivity of the lab, 3. Initial temperature = 40 4. Surface temperature = 11 5. ime of heating = 1800 10 7 m / Aumption 1. Heating from the ide i ignored. hermal diffuivity i contant
n L x 0 0.015 0 m k hl 0 F t L 10 7 m / 1800 0.015 m 0.3 i 0.0043 So the temperature = 10.65 after 30 minute of heating
Convective Boundary Condition We have conidered a negligible external fluid reitance to heat tranfer. But if we conider external fluid reitance in addition to internal fluid reitance,
At the urface, Figure 10. In convective boundary condition, urface temperature i not the ame a the bulk fluid temperature,, ignifying additional fluid reitance. k x he olution i generalized form of Eqn. 5.13 and you can refer to Heiler chart a well. h
Numerical Method a Alternative to the Chart In practice, however, uch condition dealt with above are not that imple Limitation of the analytical olution can be overcome uing numerical, computer baed olution
4 ranient Heat ranfer in a Finite Geometry Multi Dimenional Problem We hould conider the ituation two and three dimenional effect yield A finite geometry i conidered a the interection of two or three infinite geometrie (1) z lab inite i t z lab y inite i t y lab x inite i t x i t xyz inf, inf, inf,,
Figure 11. A finite cylinder can be conidered a an interec tion of an infinite cylinder and a lab lab inite i t z cylinder inite i t r i t z r inf, inf,,, ()
5 ranient Heat ranfer in a Semiinfinite Region A emi infinite region extend to infinity in two direction and a ingle identifiable urface in the other direction You can ee Fig. 5.11 extend to infinity in the y and z direction and ha an identifiable urface at x=0 Figure 1. Schematic of a emi-infinite region howing only one identifiable urface.
It can be ued practically in heat tranfer for a relatively hort time and/or in a relatively thick material he governing equation with no bulk flow and no heat generation i he boundary condition are he initial condition i t x t 0 i x 0 x i (3) (4) (5) (6)
he olution i t x erf i i 1 (7) he function erf(η) i called error function and given by t x 0 ) ( d e erf And here,
Figure 13. Comparion of the complementary error function (1-erf(η)) with an exponential e-η
Heat flux at the urface of the emi infinite region can be calculated with chain rule 0 0 " x x dx d d d k dx d k q t k t e k i i 1 0 (8)
he ituation we can approximate emi infinite region x t x 4 t (9) Figure 14. Plot of Eqn. 9, illutrating the minimum thickne of a material for which error function olution can be ued.
Other boundary condition 1. Convective boundary condition h x k urface urface he olution i k t h t x erf e t x erf k t h k hx i i 1 1 (30)
. Specified urface heat flux boundary condition " " urface q q (31) t x erf k x q e t q k t x i 1 " 4 " (3) he olution i
Example 3 Analyi of Skin Burn Figure 15. Section of a kin with degree of burn uperimpoed on it. A thermal burn occur a a reult of an elevation in tiue temperature above a threhold value for a finite period of time he intenity of thermal burn i divided into four degree
6 Chapter Summary ranient Heat Conduction No Internal Reitance, Lumped Parameter 1. he thermal reitance of the olid can be ignored if a Biot number i le than 0.1.. A thermal reitance are ignored, temperature i a function of time only.
Internal Reitance i Significant 1. When internal reitance i ignificant (Bi>0.1), temperature i a function of both poition and time. For an infinite lab, infinite cylinder and pherical geometry, the olution are given a Heiler chart. You can find it on page 37~39. 3. For finite lab and finite cylinder, the olution are interection of the infinite lab and cylinder. 4. Material with thickne L 4 t are conidered effectively emi infinite