ERT 216 HEAT & MASS TRANSFER Sem 2/ 2011-2012portal.unimap.edu.my/portal/page/portal30/Lecturer...

ERT 216 HEAT & MASS TRANSFER

Sem 2/ 2011-2012

Prepared by;

Miss Mismisuraya Meor Ahmad

School of Bioprocess Engineering

University Malaysia Perlis

Conduction 1) Steady state conduction- One

dimension

2) Steady state conduction- Multiple dimension

3) Unsteady state conduction

UNSteady-State Conduction

Concept- The Different Heat transfer problems are often

classified as being steady (steady

state) or transient (unsteady

state).

The term steady implies no

change with time at any point

within the medium, while

transient implies variation with

time or time dependence.

The temp. or heat flux remains

unchanged with time during steady

heat transfer through a medium at

any location, although both quantities

may vary from one location to

another.

Concept- situation Example: Steady State

Heat transfer through the walls of a house is steady when the

conditions inside the house and the outdoor remain constant

for several hours. But even in this case, the temp. on the inner

and outer surface of the wall will be different unless the temp.

on the inner and outer the house are the same.

Example: unsteady state (transient process)

The cooling of an apple in a refrigerator is a transient heat

transfer process since the temp. at any fixed point within the

apple will change with time during cooling.

Concept- Lumped system During transient heat transfer, the temp. normally varies with time as well

as position. In the special case of variation with time but not with position,

the temp. of the medium change uniformly with time. Such heat transfer

system are called lumped systems.

Consider a small hot copper ball coming out of

an oven. Measurement indicates that the temp. of

the copper ball change with time, but it does not

change with position. Thus the temp. of the ball

remains nearly uniform at all times. Thus lump

system analysis is applicable in this case.

Consider a large roast beef in the oven. The

temp. distribution within the roast is not even close

to being uniform. (the outer part of the roast are

well done while centre part is barely warm). Thus

lump system analysis is not applicable in this

case.

Lumped System Analysis

Consider : a body of arbitrary shape of

mass (m), volume (v), surface area

(As), density (ρ) and specific heat (Cp)

initially at a uniform temp. (Ti).

At time t=0, the body is placed into a medium at

temp. T∞, and heat transfer takes place between

the body and its environment, with a heat transfer

coefficient, h.

For discussion, we assume that T∞>Ti (but the analysis is

equal valid for the opposite case). We assume lumped

system analysis to be applicable, so that the temp.

remains uniform within the body at all times and change

with time only, T= T(t).

Lumped System Analysis During differential time interval (dt), the temp. of the body rises by a differential

amount dT. An energy balance of the solid for the time interval (dt) can

be expressed as:

Noting that and since . So that:-


Integrating from t=0 at T=Ti, to any time t at T=T(t). So that:-

Taking the exponential of both sides and rearranging, obtain:-

Unit b Where:-

So that


There are 2 observation that can be made:

1) The equation can used to determine the temp. T(t) of a body at time t

or alternatively the time t required for the temp. to reach a specified

value T(t).

2) The temp. of a body approaches the ambient temp. T∞ exponentially.

The temp. of the body change rapidly at the beginning, but rather

slowly later on. A large value of b indicates that the body

approaches the environment temp. in a short time. The larger the

value of exponent b, the higher the rate of decay in temp.


When the temp. T(t) at time t from , the rate of convection heat

transfer between the body and its environment at that time can be determine from

Newton’s Law of cooling:-

Unit

The total amount of heat transfer between the body and the surrounding medium

over the time interval t=0 to t is simply the change in the energy content of the body:-

Unit

The amount of heat transfer

reaches it upper limit when the

body reaches the surrounding

temp. T∞. There for, the maximum

heat transfer between the body

and its surrounding is:-

Unit

Criteria for Lumped System Analysis The lumped system analysis provides great convenience in heat transfer

analysis.

The 1st step in establishing a criterion for the applicability of the lumped

system analysis is to define a characteristic length, Lc and a dimensionless

Biot number. Bi.

The characteristic length, Lc to be used in the Bi no. for simple

geometries in which heat transfer is 1D, such as a large plane

wall of thickness (2L), along cylinder of radius (ro) and a sphere

of radius (ro) become L (half thickness), ro/2 and ro/3

Where

Criteria for Lumped System Analysis

Biot Number, Bi

Can expressed as:-

The Bi. no is the ratio of the internal resistance of a body to heat

conduction to its external resistance to heat convection. Therefore, a small

Bi no. represent small resistance to heat conduction and thus small

temp. gradient within the body.

OR

Criteria for Lumped System Analysis Lumped system analysis assumes a uniform temp. distribution throughout

the body, which is the case only when the conduction resistance

is zero.

The smaller the Bi no. the more accurate the

lump system analysis. It is generally accepted that lumped

system analysis is applicable if (the 2nd step)

Small bodies with high thermal conductivity are good candidates

for lumped system analysis because the Bi no. small.

Example: Lumped System Analysis Consider Small bodies with high thermal conductivities & low convection

coefficients are most likely to satisfy the criterion for lumped system analysis

So that,

Bi <0.1, satisfy the criterion for lumped system, so the lumped system

analysis can be used.

Example 1

Solution

The Transient Temperature Chart

When the lumped system analysis is not applicable, the variation of temp.

with position as well time can be determined using the transient

temp. chart (Heisler/Grober Charts) for a large plane wall, a

long cylinder, a sphere and a semi-infinite medium respectively.

Transient Heat Conduction in 1D

system

Consider: The variation of temp. with time & position in 1D problems such as those associated with a large plane

wall, a long cylinder and a sphere.

These chart are applicable for 1D heat transfer in those

geometries. Therefore, their use is limited to situation in which:-

1) the body is initially at a uniform temp.

2) All surfaces are subjected to the same thermal conditions, and

3) The body does not involve any heat generation.

These charts can also be used to determine the total

heat transfer from the body up to a specified time, t.



Schematic of the simple geometries in which heat transfer in 1D

Plane wall Cylinder

Sphere

The Transient Temperature Chart (PLATE)

Transient temp. & heat transfer charts for a plane wall of thickness 2L initially at a

uniform temp. Ti subjected to convection from both sides to an environment T∞ with a

convection coefficient of h. The height & the width of the wall are large relatively to its

thickness, and thus heat conduction in the wall can be approximated to be 1D. Also,

there is thermal symmetry about the midplane passing through x=0, & thus the temp.

distribution must be symmetrical about the midplane. Therefore, the value of temp. at

any –x value in –L ≤ x ≤ 0 at any time must be equal to the value at +x in 0 ≤ x ≤ L at

the same time.

This means we can formulate & solve the heat conduction problem in the

positive half domain 0 ≤ x ≤ L and the apply the solution to the other half.

Situation:-

Midplane temperature

Temperature Distribution

Heat Transfer

Transient temp. & heat transfer charts for a long cylinder of

radius ro initially at a uniform temp. Ti subjected to

convection from all sides to an environment T∞ with a

convection coefficient of h.

The Transient Temperature Chart (CYLINDER)

Heat Transfer

Transient temp. & heat transfer charts for a sphere of radius

ro initially at a uniform temp. Ti subjected to convection from

all sides to an environment T∞ with a convection coefficient

of h.

The Transient Temperature Chart (SPHERE)

Heat Transfer

Solution of 1D transient heat conduction problems

Using the one-term approximation, the solution of 1D transient

heat conduction problems are expressed analytically as:

Where the constant A1 and λ1 are functions of the Bi no. only,

and their value are listed in table against the Bi no. for all three

geometries.

The error involved in one-term approximation is less than 2%

when Fo or >0.2


Using the one-term solutions, the fractional heat transfers in different

geometries are expressed as:

Solution of 1D transient heat conduction in semi-infinite solid

The solution of transient heat conduction in a semi-infinite solid with constant

properties under various boundary condition at the surface are given as follow:-

1) Specified Surface Temp., Ts= constant

2) Specified Surface Heat Flux, qs= constant

3) Convection on the Surface, qs(t) = h[T∞- T(0,t)]

4) Energy Pulse at surface, es= constant

Where erfc(ɳ) is the complementary error function of argument ɳ.

Summary Summary of the solutions for 1D transient conduction in plane

wall of thickness 2L, a cylinder of radius, ro and a sphere of

radius ro subjected to convection from all surface

Coefficient used in the one-

term approximate solution of

transient 1D heat conduction

in plane walls, cylinders and

sphere (Bi = hL/k for a plane

wall of thickness, 2L and Bi

=hro/k for a cylinder or

sphere of radius ro)

The zeroth- and first order Bessel

functions of the first kind

Term- transient heat conduction Bi no. measure of the relative magnitudes of

the 2 heat transfer mechanism: convection at the

surface & conduction through the solid. A small

value of Bi no. indicates that the inner resistance

of the body to heat conduction is small relative to

the resistance to convection between the surface

& the fluid. As a result temp. distribution within

the solid become fairly uniform and lumped system

analysis become applicable (Bi<0.1)

Fourier no. (ד or Fo) measured of heat conducted through a body relative

to heat stored. Thus, a large value of the Fourier no. indicates faster

propagation of heat through a body.

Example 2

Solution

Solution Using the one-term approximation

Example 3

Solution

Example 4

Solution


Using a superposition principle called the product solutions, the charts (the

transient temp. chart used to determine the temp. distribution & heat transfer

in 1D heat conduction problem) can also be used to construct solutions for the

2D transient heat conduction problems encountered in geometries (a

short cylinder, a long rectangular bar or a semi-ifinite cylinder or plate)

even 3D problem associated with geometries (rectangular prism or a

semi-infinite rectangular bar), provided that all surfaces of the solid are

subjected to convection to same fluid at temp. T∞ with the same convection

heat transfer coefficient, h & the body involved no heat generation.

Transient Heat Conduction in

Multidimensional Systems

The solution in such multidimensional geometries can be expressed as the

product of solution for the 1D geometries whose intersection is the

multidimensional geometry.


The total heat transfer to or from a multidimensional geometry

can also be determined by using the 1D value. The transient

heat transfer for 2D geometry formed by the intersection of two

1D geometries 1 & 2 is:

Transient Heat Conduction in

Multidimensional Systems

Transient heat transfer for 3D body formed by the intersection

of three 1D bodies1, 2 & 3 is given by:

Most heat transfer problems encountered in practice are

transient (unsteady state) in nature, but they are usually

analyzed under some presumed steady conditions since steady

processes are easier to analyze and they provide the answers

to our questions.

Conclusion

Unsteady-state heat transfer analysis is obviously of

significantly practical interest because of the large no.

of heating and cooling process that must be

calculated in industrial applications.

To analyze transient heat-transfer problem solving the

general heat-conduction equation by separation-of-

variables method (similar to the analytical treatment used

for the 2D steady state problem).

Conclusion Example:

Heat transfer through the walls & ceiling of a typical house is never

steady since the outdoor conditions such as the temp. the speed and

direction of the wind, the location of the sun and so on, change

constantly. The conditions in a typical house are not so steady either.

Therefore, it is almost impossible to perform a heat transfer analysis of

the house accurately. That’s why we really need heat transfer analysis.

If the purpose of a heat transfer analysis of a house is to determine

the proper size of heater, so that, we need to know the max. rate of

heat loss from the house, which is determine by considering in heat

loss from the house during steady state operation under worst

conditions. Therefore, we can get the answer to our question by doing

a heat transfer analysis under steady conditions.

Tutorial 4

Book: J.P. Holman

1) 4-5

2) 4-11

3) 4-13

4) 4-15

5) 4-26

6) 4-40

ERT 216 HEAT & MASS TRANSFER Sem 2/ 2011-2012portal.unimap.edu.my/portal/page/portal30/Lecturer...

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Transcript of ERT 216 HEAT & MASS TRANSFER Sem 2/ 2011-2012portal.unimap.edu.my/portal/page/portal30/Lecturer...