Heat and Mass Transfer

이 윤 우

서울대학교 화학생물공학부

19UNSTEADY-STATE

HEAT CONDUCTION

Supercritical Fluid Process Lab

Unsteady-State Heat Conduction

is of importance to engineers in many circumstances

It may control the rate at which process equipment is brought to stable operating conditions

It is also important in determining the processing time of many solid articles. For example, the curing time of objects made of molded plastic or rubber is often dependent on the time required to bring the center to some specifiedtemperature without causing thermal damage to the material at the surface.

There are also many applications of unsteady-state-conduction theory in the heat treating and casting of the metals.

A hot metal billet that is removed from a furnace

Re-entry of Columbia into the earth’s atmosphere

200oC 40oC

θ=20min

200oC 40oC

θ=60min

q1 = q2

200oC 40oC

θ=20min

100oC 50oC

θ=60min

q1 ≠ q2

(a) Steady (b) Unsteady

Steady vs. Unsteady State Heat Transfer

Unsteady problem typically arise when the boundary conditions of a system are changed.If the surface temperature of a system is altered, the temperature at each point in the systemwill also begin to change. The changes will continue to occur until a new steady-state temperature distribution is reached.

The first law of thermodynamics

Ein Eout

Control volumeEgenEacc

Ein – Eout + Egen = Eacc

Ein : Energy transport by the fluid into the control volume surface phenomenonEout : Energy transport by the fluid out of the control volume surface phenomenonEgen : Energy generation in the control volume volumetric phenomenonEacc : Energy accumulation in the control volume volumetric phenomenon

re temperatu:T time,:

heat Specific:C volume,:V density,:

volume:

eunit volumper rate generationheat :

∂∂

qx+Δx

qz+Δz qy+Δy

General heat conduction equation

Energy Balance: In - Out + Generation = AccumulationEin - Eout + Egen =Eacc

Egen = gV (g: heat generation rate/volume)

Eacc = )( VTCpρθ∂∂

dxdydzgEztdxdykq

ytdxdzkq

xtdydzkq

&=∂∂

∂∂

⋅⋅∂=

∂∂

)( tdxdydzCE

dxxqqq

General heat conduction equation

tCgztk

∂∂

=+∂∂

+∂∂

∂∂

=+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

For constant properties (k, ρ, Cp)

: thermal diffusivity : thermal capacitance

Materials of large α will respond quickly to changes in their environment, taking shorter to reach a new equilibrium condition.

Fundamental Equations

∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

+∂∂

ρθ(8-11)

(8-12)

(19-1)

The basic differential equation for unsteady-state conduction in a solid(For constant thermal conductivity and no heat generation)

Unsteady-state conduction in only x direction

A differential energy balance

=∂∂

+∂∂

(19-2)

Steady-state conduction in only x direction

Laplace’s equation

The basic differential equation for steady-state conduction in a solid(For constant thermal conductivity and no heat generation)

Which is readily integrated to give a linear relation between temperature and distance. This was shown earlier in Eq. (18-3), obtained by integrating the Fourier conduction equation.

General Heat Transfer Equation

Cylindrical coordinates

zzryrx

sincos

zrrV ΔΔΔ=Δ θφrd

rΔzΔ

θΔθ

θθρ

θθθθθθ

′∂∂

ΔΔΔ

ΔΔΔ∂∂

ΔΔ−−∂∂

ΔΔ−=−

∂∂

ΔΔ−−∂∂

ΔΔ−=−

∂∂

ΔΔ−−∂∂

ΔΔ−=−

Δ+Δ+

rzrSztzkr

ztrkrqq

rtzrkqq

rtzkrqq

zzzzzz

rrrrrr

Input-Output

r-direction:

θ-direction:

z-direction:

Generation:

Accumulation:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=′∂

′∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

′∂∂

∂∂

−∂∂

∂∂

−∂∂

∂∂

−∂∂

→ΔΔΔ

Δ+Δ+Δ+

rrrtkr

pzzzrrr

θρθ

θθθ

If k=constant

)( rzr ΔΔΔ÷ θ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=′∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=′∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=′∂

θρθ

θρθ(19-3)

No axial conduction

The basic differential equation for unsteady-state conduction in a solid(For constant thermal conductivity and no heat generation)

No axial conduction & angular conduction

RadialConduction

AxialConduction

θφθφθ

cossinsincossin

zryrxθΔr

φθ Δ⋅sinr

Δφ)Δθ, φΔr, θ(r +++

),,( φθr

rΔrrV Δ⋅Δ⋅Δ⋅=Δ θφθsin2

Spherical coordinates

θφθθρ

φθθ

θφθ

φθθφθθ

φφφ

θθθθθθ

′∂∂

ΔΔΔ

∂∂

ΔΔ−−∂∂

ΔΔ−=−

∂∂

ΔΔ−−∂∂

ΔΔ−=−

∂∂

ΔΔ−−∂∂

ΔΔ−=−

Δ+Δ+

rtrkrqq

rtkrqq

rrrrrr

sin1)(

)sin()sin(

Input-Output

r-direction:

θ-direction:

z-direction:

Generation:

Accumulation:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=′∂

′∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

′∂∂

∂∂

−∂∂

∂∂

−∂∂

∂∂

−∂∂

→ΔΔΔ

Δ+Δ+Δ+

sin1sin

sin11lim

sinsinsin

sinsin

φθθθ

θθρθ

φφθθθ

θφθφφ

θθθ

φφφθθθ

rrrtkr

If k=constant

)sin( 2 φθθ ΔΔΔ÷ rr

Differential Energy Balance

&+⎥⎦

⎤⎢⎣

⎡∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+′∂

2 sin1sin

φθθθ

θθρ

φθθθφθ

pzr &+⎥

⎤⎢⎣

⎡∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=∂∂

+∂∂

+′∂

11θρθθ

pzyx &+⎟⎟

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

+∂∂

Rectangular coordinates

Energy Equations by ConductionConstant Properties

&+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=′∂

2 sin1sin

φθθθ

θθρθ

&+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=′∂

11θρθ

&+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

Rectangular coordinates

One-Dimensional Conduction in a Planar Medium with Constant Properties and No Generation

&+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

p ∂∂

=∂∂

pzyx &+⎟⎟

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

+∂∂

Boundary and Initial Conditions

Constant Surface Temperature

Constant Heat Flux

Applied Flux

Insulated Surface Convection

p ∂∂

=∂∂

For transient conduction, heat equation is first order in time, requiring specification of an initial temperature distribution:

Since heat equation is second order in space, two boundary conditions must be specified. Some common cases:

),( θxt ),( θxt),( θxt

),( θxt

),0( θt

Boundary Conditions at the surface (x=0)

1. Constant Surface Temperature

stt =),0( θst

),( θxt

2. Constant Surface Heat Flux(a) Finite heat flux

Applied Flux

),( θxt

qxtk ′′=∂∂

2. Constant Surface Heat Flux(b) Adiabatic or insulated surface

=∂∂

),( θxt

3. Convection Surface Condition

),( θxt

),0( θt

ht ,∞

( )[ ]θ,00

tthxtk

−=∂∂

− ∞=

Ex. 19-1Problems of heating or cooling spheres are best solved using the differential energy balance written in spherical coordinates. Derive the equation for the case where there is no variation of temperature with angular position.

Control volume

Rate of heat flow into control volume

Rate of heat flow out of control volume

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

++−=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

+∂∂

∂∂

drrtrdr

drrtdrdr

rtrdrdr

rtdrrdrrk

rtdrrk

Net heat flow000

spherical coordinates

Ex. 19-1

Control volume

The rate of accumulation energy in the control volume

Net heat flow = rate of accumulation

⎟⎟⎠

⎞⎜⎜⎝

∂∂

+∂∂

θ∂∂

ρπ=⎟⎟⎠

⎞⎜⎜⎝

∂∂

+∂∂

θ∂∂

tCdrrdrrtrdr

spherical coordinates

The equation for unsteady conduction in a sphere

Modeling in Heat Transfer

Potato

PhysicalPhenomena

Actual Ideal(simplification)

Modeling

PhysicalPrinciples & laws

∂∂

=∂∂ αθ

ODE, PDE

Mathematicalformulation

Solution

Physical PhenomenaPhysical Phenomena

Solution of the Fundamental Equations

ODE, PDEODE, PDE

Numerical SolutionNumerical Solution

Modeling

Analytical SolutionAnalytical Solution

(1) Separation of variables(2) Similarity variable(3) Laplace, Fourier transform

- FDM, FEM

Similarity Variable

Conduction into a plate of infinite thickness

Analytical Solution

t t,at x :2 B.C.t t0,at x :1 B.C.t t0,θat : I.C.

=∞=====

∂∂

=∂∂

(19-1)

(19-4)

Similarity VariableAnalytical Solution

xnnxxn

∂∂

⋅=∂∂⋅

∂∂

⋅=∂∂⋅⎟⎠

⎞⎜⎝

⎛∂∂

⋅∂∂

=∂∂

∂∂

⋅=∂∂⋅

∂∂

=∂∂

∂∂

⋅−=∂∂

⋅∂∂

=∂∂

−=⋅−=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∂∂ −

αθαθαθ

θθθ

αθθθαθθ

∂∂

⋅⋅=∂∂

⋅−

∂∂

⋅=∂∂

∂∂

⋅−=∂∂

αθα

θθ (19-5)

(19-6)

(19-7)

: variablenew

CdneCt

npdndp

∫ −

(19-8)

(19-9)

(19-10)

(19-7)

Two boundary conditions are needed. Supercritical Fluid Process Lab

=+dndtn

αθπ

,n 0, , x:2 B.C.C ,t t0,n 0, x:1 B.C.

xerfdnetttt

tdneCt

==−−

=∞→=∞→====

∞ −(19-11)

(19-12)

(19-13)

Gauss error function or probability functionSupercritical Fluid Process Lab

The error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. One of its main purposes is for graphing a normal distribution. It is defined as:

By expanding the right-hand side in a Taylor series and integrating, one can express it in the form

for every real number x, or any complex number z = x. The error function is evidently odd.

Plot of the error function

Analytical Solution

αθπαθ

2 xerfdnetttt x

s ==−−

∫ −

Gauss error function (probability function)

erf(z)

zπ2erf(z) z, smallfor =

Analytical Solution

αθαθπ

αθπαθ

018.0~42

==−−

∫ −

xerfdnetttt

Thermal penetration depth, xT

x @ 1% of driving force

zπ2erf(z) z, smallfor =

Analytical Solution

smallisxxxerftttt

−⎟⎟

⎜⎜

−⎟⎠

⎞⎜⎝

⎛ −−=

∂∂

≈=−−

παθ

αθπαθ

x @ 1% of driving force

pCkρ : Thermal effusivity

Analytical Solution

과도기 상태에서는 가 중요하다.pCkρ

겨울에 나무 손잡이와 쇠 손잡이를 잡을 때더 빨리 차가움을 느끼는 것

뜨거운 공기와 뜨거운 물에 손을 담글 때화상을 입는 정도

100oC 스팀 사우나에서는 오랫동안 견디면서45oC 목욕탕 물 속에서는 잠깐도 견디기 어려운 것

Separation of Variables

Unsteady state heat conduction in a rubber sheet

Analytical Solution

/0028.0Ck

F290 tand 0 x@ ?θ

0dt/dx 0,at x:2 B.C.

F292t t,1/4"xat x:1 B.C.

F70t t0,θat : I.C.

∂∂

α=θ∂∂

(19-1)

x0=1/4”

Curing at 292oFfor 50min

Analytical Solution

1)(1)(

ttYYtt

∂∂

−=∂∂

∂∂

−=∂∂

∂∂

−=∂∂

∂∂

=∂∂

∂∂

−=∂∂

∂∂

=∂∂

==−−

∂∂

=∂∂

αθτ

αθ (19-1)

Dimensionless variables

Analytical Solution

0,1,:2..

0,00,0:1..

1,0,0:..

==→==

=∂∂

=→=∂∂

==→==∂∂

=∂∂

∂∂

−=∂∂

∂∂

=∂∂

YnttxxCBnYn

YttCInYY

τθτ

αθ (19-1)

(1-1) & (1-2) (19-1)

Analytical Solution

0,0@:B.C.2

0,1@:B.C.11,0@:I.C.

=∂∂

∂∂

=∂∂

0dt/dx 0, x@:2 B.C.t t,x x@:1 B.C.

tt0,θ @: I.C.

∂∂

=∂∂

xtt α

xnttttY

s αθτ ==−−

Dimensionless variables

Analytical Solution

=∂∂

τSolution: (3)

(5)Supercritical Fluid Process Lab

Separation of variables:

Separation of VariablesT is a function of only τN is a function of only n

∂∂

=∂∂τ

Left is a function of only τRight is a function of only n

Analytical Solution

( )anCanCeCNTY

anCanCNNadn

eCTTaddT

cossin

cossin0

+=⋅=

+=→=+

=→=+

(6) (8)

(7) (9)

4 unknowns

Since time and distance are independent of each other. The left side of this equation is independent of distance and the right side is independent of time. Hence each side must be constant.

Analytical Solution

ππππ

212,2/5,2/3,2/

cos0:2..

)A (wherecos

0sincos:1..

0,1,:2..

0,00,0:1..

1,0,0:..

==−=∂∂

==→==

=∂∂

=→=∂∂

==→==

−−

CCanAeY

eCaCanaCanaCeCnYCB

YnttxxCBnYn

This equation contains four constants C1, C2, C3, and a: however C1 can be combined with C2 and C3, so that only three IC & BCs are needed to complete the solution.

n=0, sinan=0, cosan=1

(symmetrical system)

Analytical Solution

+⎟⎠⎞

⎜⎝⎛ −

+⎟⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

−−

neAneAY

)12(cos

)2/)12[(

)2/5(3

)2/3(2

τπτπ

GeneralSolution

The substitution of one of these values for a in Eq. (11) would meet the requirement of the second boundary condition, but it is not possible to represent an arbitrary temperature distribution in the slab by a cosine curve.

Analytical Solution

LL +⎟⎠⎞

⎜⎝⎛ −

++⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= niAnAnA i 2

)12(cos2

cos1 21πππ

Both side of this equation are multiplied byAnd integrated over the range of 0 to 1.

ni π⎟⎠⎞

⎜⎝⎛ −

2)12(cos

idnni )1(1

212sin1

212cos

−−=⎥

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−=⎟

⎠⎞

⎜⎝⎛ −

∫ ππ

( ) 02

sin)22(1sin

)22(1sin

212cos

=⎥⎦

⎤⎢⎣

−−

⎥⎦

⎤⎢⎣

−−

=⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛

ππππ

iniAdnninA

Analytical Solution

LL +⎟⎠⎞

⎜⎝⎛ −

++⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

)12(cos2

cos1 21πππ

Both side of this equation are multiplied byAnd integrated over the range of 0 to 1.

ni π⎟⎠⎞

⎜⎝⎛ −

2)12(cos

212)2sin(

212cos

=⎥⎦⎤

⎢⎣⎡ −−+

−−

⎥⎦⎤

⎢⎣⎡ −

⎟⎠⎞

⎜⎝⎛ −

πππ

Analytical Solution

LL +⎟⎠⎞

⎜⎝⎛ −

++⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

)12(cos2

cos1 21πππ

(13)I.C.

−−

= 0 + 0 + 0 + … + 2iA

)12()1(4

321 πππ

=−==

−−−

=−−

Analytical Solution

[ ] niei

⎟⎠⎞

⎜⎝⎛ −

−−−

−⎟⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

−−∞=

−−

∑ 2)12(cos

2/)12()1(2

]2/)12[(

)2/3()2/(

τπτπ

The general solution is

Analytical Solution

L−π

−−

τπ−τπ−τπ− 222 )2/5()2/3()2/(

344009.0

0@70292

290292

The specific example:

The solution for τ must be obtained by trial and error.As a first approximation, only the first term on the right hand side will be considered. This gives

min)7.18(313.0)01.2(121

0028.011

)009.0)((ln2

C p =⎟⎠⎞

⎜⎝⎛=τ

⎟⎠⎞

⎜⎝⎛π

−=τ

(18.7min)

Analytical Solution

)1040.1)(254.0()1050.3)(424.0()00694.0)(27.1(

1248.4497.4

−−

−−−

×+×−=

Check the relative magnitude of the terms in the series solution

The validity of the approximation which employed only the first term of the series is apparent.

Analytical Solution

Temperature distribution in rubber sheet

0 1 70 70 70 70

1 9.76x10-1 75 92 139 209 292

5 3.33x10-1 217 223 239 263 292

10 9.02x10-2 272 273 278 284.3 292

20 6.37x10-3 290.6 290.7 291 291.5 292

30 4.52x10-4 291.9 291.9 292 292 292

40 3.19x10-5 292 292 292 292 292

TimeElapsed(min)

π2)2/(4 −e

Temperature in sheet, oF

⎟⎠⎞

⎜⎝⎛

924.02

⎟⎠⎞

⎜⎝⎛

707.02

⎟⎠⎞

⎜⎝⎛

383.02

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

Analytical Solution

Temperature profiles in rubber sheet

00 1/4 1/2 3/4 1surfacecenter

Distance, n(=x/x0)Supercritical Fluid Process Lab

Analytical Solution Conduction in Cylinder

sttRrrtrat

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=′∂

0 0 (L>>R)

B.C.1:

B.C.2:

0=∂∂rt

Analytical Solution Conduction in Cylinder

0,0 =∂∂

sttRr == ,0,0 tt ==θ

),( θ= rft

Conduction in CylinderAnalytical Solution

rNdrNd

−==+

∂∂

=∂∂

+∂∂

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=′∂

Solution:

)exp(0

−=→=+

−==+

NardrdNr

aCTTaddT

rNdrNd

θααθ

∞ −−

⎟⎠⎞

⎜⎝⎛−

=−++

)1(!21)1(

xBYxAJy

ykxdxdyx

Bessel’s equation of order k

Solution:

constant sEuler' :5772.0ln131

211lim

)!(21)1(

2)(21ln2)(

)1(!21)1(

LL =⎟⎠⎞

⎜⎝⎛ −++++=

⎟⎠⎞

⎜⎝⎛−

−⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛−

==→=++

∞→

ararJararY

arBYarAJN

arxandkNardrdNr

Bessel function of the 1st kind of order zero

Bessel function of the 2nd kind of order zero

0,0:2..

0,:1..1,0:..

)()()()( 001

=∂∂

+=⋅= −

YRrCBYCI

arBYarAJecrNTY a

θ θα

differentiation

)()()(

xxYxkYxYdydx

xxJxkJxJdydx

αααα

)()()()(

araYarYdrdararYarY

araJarJdrdararJarJ

−=→−=

0,0@ =∂∂

))()()(exp( 112

1 arBaYarAaJacrY

+−−=∂∂ θα

))0()0()(exp( 112

10 BaYAaJacrY

r +−−=∂∂

= θα

001 =→=∴ BBc

Supercritical Fluid Process Lab( ))()()()( 001

arBYarAJecrNTY a +=⋅= − θαθ

0,@ == YRr

)()exp(0 02

1 aRJaAc θα−=

0)( ofroot th theis a 0 =RaJ iιι

∑∞

02 )()exp(

iiii raJacY θα

Supercritical Fluid Process Lab( ))()()()( 001

arBYarAJecrNTY a +=⋅= − θαθ

1,0@ == Yθ

)(1 00

raJc ii

i∑∞

integralthen),(0 raxJa jj×

)()( 1,kwhen

function :)()1()(

xJxdxxJx

ααα

Γ−=

−∫

Left: )()( 10 1 RaRJdxxJx j

kk =∫ − αα

)()( 10 0 RaRJdrraxJa j

jj =∫

Supercritical Fluid Process Lab∑∞

02 )()exp(

iiii raJacY θα

right: ji ≠

∫ =−−

jijijiji

ijii RaJRaJaRaJRaJa

aaRCdxxaJxaxJC

0 10102200 0)]()()()([)()(

∫ −−=R

ii RaJRaJRaJRaCdxxaxJC0 11

20 )]()()([

)]()([2

)( 111 RaJRaJRaCRaRJ iiii

j −−=

11 RaJRaRaJRaC

iiiii =

−=∴

)/exp(2

)()exp()(

raJRaJa

raJaRaJRatt

iii iis

−=−−

Final Solution

)/exp(20

raJRaJa

CkaRtt

s ∑∞

−− ρθ

Final Solution

0)( ofroot th theis a 0 =RaJ iιι

J0 and J1: Bessel function of the 1st kind (zero and first order)

(19-15)

The Fourier equation has been solved for many geometries and sets of conditions. A set of general solutions has been plotted for use in obtaining reasonably good solutions with less work. These are the Gurney-Lurie or Heisler Charts. Figures 19-3, 19-4, and 19-5 are simple versions of such a chart.

Use of the charts is restricted to cases where: 1. there is no internal heat source (generation) 2. the thermal diffusivity of the object is constant 3. the problem can be approximated as one-dimensional 4. the initial temperature of the object is uniform 5. the system is forced by a step change in temperature

of the surroundings (or of the surface, when 1/h=0)

Graphical Solution: Gurney-Lurie Chart

Graphical Solution: Gurney-Lurie ChartThe chart below is only an example of how a Gurney-Lurie chart might look and is not based on any actual data. The chart shows how four different dimensionless groups depend on each other.

Bihxkm

s 1,,,00

=====−−

θαθ

0ttttY

−−

Xpρθαθ

Gurney-Lurie Chart

To use the charts, some variables need to be defined. Various versions of the charts are slightly different in how the variables are defined:

resistance relative :1

position relative :

time)(relativenumber Fourier :

Bihxkm

=−−

ρθαθ

change tempshedunaccompli fractional the:

Fourier number (Fo)

Supercritical Fluid Process Labstored

conducted

Gurney-Lurie Chart for Plate

Gurney-Lurie Chart for Cylinder

Gurney-Lurie Chart for Sphere

Gurney-LurieChart for Sphere

Gurney-Lurie Chart for Sphere

Newman’s Rule :the technique for solving many problems of systems with finite dimensions in all direction

Finite dimension in all direction

If a brick-shape object is heated or cooled, the general solution describing temperature as a function of time and three distance variables, x, y, and z.

Y=YxYyYz

Yx=f1(x, θ) : x-dir. unsteady state conductionYy=f2(y, θ) : y-dir. unsteady state conductionYz=f3(z, θ) : z-dir. unsteady state conduction

009.070292

29029270292

=−−

−−

=−−

tttttY

t = 290oF at center θ = ?

min)7(113.044.673.0

48/10028.044.673.0

3-19 Fig. From.0,0

208.00090.0

⎟⎟⎠

⎞⎜⎜⎝

⎛ θ=θ==

Newman’s Rule: Y=YxYyYz

009.070292

29029270292

=−−

−−

=−−

tttttY

h→∞

Three dimensional conductionThree dimensional conduction

p ∂∂

+∂∂

=∂∂

→⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

τρθ

1111zY

YYxYYYYYY

xzyzyx

∂∂

=∂∂

∂∂

=∂∂

τττττ

)()()( zYyYxYY zyx=τ

The basis for validity of Newman’s Rule

Solution:

Differentiating of Eq. (2)

Rearrangement yields

++−=∂∂

−=∂∂

−=∂

−=∂∂

Newman’s RuleEach part of Eq(3) is a function of one of the four independent variables and hence, it is reasoned, each of the four terms is a constant.

( ) ( )( )( )zaCzaCyaCyaC

xaCxaCeCY

zaCzaCY

yaCyaCYxaCxaCY

37362524

cossincossincossin

cossin

cossincossin

Newman’s Rule

These individual solutions are combined

These four expressions can be written as ordinary differential equationsfor which the solutions are

These general solution for three-dimensional conduction is consist of the products of each of the one-dimensional solutions.

Homework #2

PROBLEMSPROBLEMS

1919--111919--22

Due date: October 19Due date: October 19

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