Transcript HEAT PROCESSES

HP4
HEAT PROCESSES
Heat transfer
Mechanisms of heat transfer. Conduction, convection (heat transfer coefficients),
radiation (example: cooling cabinet). Fourier’s law of conduction, thermal resistance
(composed wall, cylinder). Unsteady heat transfer, penetration depth (derivation, small
experiment with gas lighter and copper wire). Biot number (example: boiling potatoes).
Convective heat transfer, heat transfer coefficient and thickness of thermal boundary
layer. Heat transfer in a circular pipe at laminar flow (derivation Leveque). Criteria: Nu,
Re, Pr, Pe, Gz. Heat transfer in turbulent flow, Moody’s diagram. Effects of variable
properties (Sieder Tate correction for temperature dependent viscosity, mixed and
Rudolf Žitný, Ústav procesní a
natural convection).
zpracovatelské
techniky ČVUT FS 2010
HP4
Mechanisms of heat transfer
There exist 3 basic mechanisms of heat transfer between
different bodies (or inside a continuous body)
Conduction in solids or stagnant fluids
Convection inside moving fluids, but first of all we shall
discuss heat transfer from flowing fluid to a solid wall
Radiation (electromagnetric waves) the only mechanism
of energy transfer in an empty space
Aim of analysis is to find out relationships between heat flows
(heat fluxes) and driving forces (temperature differences)
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Heat flux and conduction
General form of transport
Fourier equation
for temperature field T(t,x,y,z) in a solid or in a stagnant fluid
taking into account internal heat sources and an adiabatic
temperature increase during compression of gas.
cp
T
t
accum ulation
of enthalpy

p
t
only for gas
(com pressive
heating).
   (  T ) 
heat flux
conduction
Q
(R)
reaction heat,
phase change enthalpy
and electric heat
Benton
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Heat flux and conduction
Energy balance of a closed system dq = du + dw (heat delivered to system equals
internal energy increase plus mechanical work done by system) tells nothing
about intensity of heat transfer at the surface of system, neither about
relationships between heat fluxes and driving forces (temperature gradients). This
problem is a subject of irreversible thermodynamics.

n
dA

q
Intenzity of heat transfer through element dA of
boundary is characterized by vector of heat flux

[W/m2] q
Direction and magnitude of heat flux is determined by
gradient of temperature and thermal conductivity of
media 

q  T
Fourier’s law of heat
conduction
 
Heat flow through boundary is projection of heat flux to the outer normal n  q
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Thermal conductivity 
Thermal and electrical conductivities are similar: they are large for metals (electron
conductivity) and small for organic materials. Temperature diffusivity a is closely

related with the thermal conductivity
a 
c p
Memorize some typical values:
Material
 [W/(m.K)]
a [m2/s]
Aluminium Al
200
80E-6
Carbon steel
50
14E-6
Stainless steel
15
4E-6
Glas
0.8
0.35E-6
Water
0.6
0.14E-6
Polyethylen
0.4
0.16E-6
Air
0.025
20E-6
Thermal conductivity of nonmetals and gases increases with temperature (by
about 10% at heating by 100K), at liquids and metals  usually decreases.
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Conduction – Fourier equation
Distribution of temperatures and heat fluxes in a solid can be expressed in differential
form, based upon enthalpy balancing of infinitesimal volume dv

h
t
Accumulation of
enthalpy in unit
control volume
   q
q x
 q 
x

q y
y

q z
z
Divergence of heat fluxes (positive
if heat flows out from the control
volume at the point x,y,z)
Integrating this differential equation in a finite volume V the integral enthalpy balance
can be expressed in the following form using Gauss theorem


n
dA

q
h
V
Accumulation of
enthalpy in
volume V
t
dv      qdv    n  qds
V
S
Heat transferred
through the
whole surface S
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Conduction – Fourier equation
Heat flux q as well as the enthalpy h can be expressed in terms of temperatures,
giving partial differential equation – Fourier equation

h
t
cp
cp
Internal heat source (e.g.
enthalpy change of a chemical
reaction or a volumetric heat
produced by passing electric
current or absorbed
microwaves)
   q  Qg
T
t
T
t
   (  T )  Q g


x
(
T
x
)

y
(
T
y
)

z
(
T
z
)  Qg
Thermal conductivity  need not be
a constant. It usually depends on
temperature, and for anisotropic
materials (e.g. wood) it depends
also on directions x,y,z – in this
case ij should be considered as the
second order tensor.
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Conduction - stationary
Let us consider special case: Solid homogeneous body (constant thermal conductivity
and without internal heat sources). Fourier equation for steady state reduces to the
Laplace equation for T(x,y,z)
 T
 T
2
0   T 
2
x
 T
2
2
0
0
x
2

1 
r r
2
2

y
1 
r r
(r
2
(r
T
r
T
r
2
)
 T
2

z
2
The same equation written in
cylindrical and spherical
coordinate system (assuming
axial symmetry)
)
Boundary conditions: at each point of surface must be prescribed either temperature
T or the heat flux (for example q=0 at an insulated surface).
Solution of T(x,y,z) can be found for simple geometries in an analytical form (see next
slide) or numerically (using finite difference method, finite element,…) for more
complicated geometry.
Example temperature profile in a cylinder
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Calculate radial temperature profile in a cylinder and sphere (fixed
temperatures T1 T2 at inner and outer surface)
R1
R2
cylinder
r
r
T
r
2
T
r
 c1 ,
 c1 ,
Sphere (bubble)
T = c 1 ln r  c 2
c1 
c1
c1 
T=-
r
 c2
T 2  T1
ln R 2 / R1
R1 R 2
R 2  R1
c2 
T1 ln R 2  T 2 ln R1
ln R 2 / R1
( T 2  T1 ) c 2 
T 2 R 2  T1 R1
R 2  R1
Conduction – thermal resistance
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Knowing temperature field and thermal conductivity  it is possible to calculate heat
fluxes and total thermal power Q transferred between two surfaces with different (but
constant) temperatures T1 a T2
T1
T2
T1 T2
S2
R2
S
Q
S1
Q 
T2
R1
R1
T1
h
h1 h2
1 h1 h 2
( 
)
S 1  2
Serial
RT [K/W] thermal resistance
L
L
RT 
RT
h
T1 T2
RT 
h
S 1 1  S 2  2
Parallel
RT 
ln R 2 / R1
2  L
Tube wall
RT 
T1  T 2
ln 2 h / R
2  L
Pipe burried under surface
In this way it is
possible to express
thermal resistance
of windows, walls,
heat transfer
surfaces …
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Conduction - nonstacionary
Time development of temperature field T(t,x,y,z) in a homogeneous solid body without
internal heat sources is described by Fourier equation
T
t
 T
2
 a(
x
2
 T
2

y
2
 T
2

z
2
)
The coefficient of temperature diffusivity a=/cp is the ratio of
temperature conductivity and thermal inertia
with the boundary conditions of the same kind as in the steady state case and with
initial conditions (temperature distribution at time t=0).
This solution T(t,x,y,z) can be expressed for simple geometries in an analytical form
(heating brick, plate, cylinder, sphere) or numerically.
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Theory of penetration depth
Development of temperature profile in a half-space. Use the acceptable
approximation by linear temperature profile

Tw

T
0

t
T0
t
t+t
x

t
t
dx   a

 Tdx  a
0
(
Tw
2
)a
T
x
|x  0
Integrate Fourier equations
(up to this step it is
accurate)
Tw

Tw
Approximate
temperature
profile by line

δ
+Δ
Result is ODE for thickness  as a function of time
 
4at
Using the exact temperature
profile predicted by erffunction, the penetration
depth slightly differs =(at)
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Theory of penetration depth
=at penetration depth. Extremely simple and important result, it
gives us prediction how far the temperature change penetrates at the time t. This
estimate enables prediction of thermal and momentum boundary layers thickness
etc. The same formula can be used for calculation of penetration depth in
diffusion, replacing temperature diffusivity a by diffusion coefficient DA .
Wire Cu
=0.11 m
=398 W/m/K
=8930 kg/m3
Cp=386 J/kg/K

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Convection
General form of transport Fourier Kirchhoff equation
cp (
T
t
 u T ) 
co n vectio n
Dp
Dt
th is term is zero fo r
in co m p ressib le liq u id
  q  :  Q
co n d u ctio n
(R)
visco u s
h eatin g
Benton
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Convection
Calculation of heat flux q from flowing fluid to a solid surface requires calculation of
temperature profile in the vicinity of surface (for example temperature gradients in
attached bubbles during boiling, all details of thermal boundary layer,…).
Engineering approach simplifies the problem by introducing the idea of stagnant
homogeneous layer of fluid, having an equivalent thermal resistance (characterized
by the heat transfer coefficient  [W/(m2K)])
f
T
q   (T f  T w )   f

(T f  T w )
q   (T f  T w )
y

Tf is temperature of fluid far from surface (behind the boundary of thermal boundary layer), Tw is wall
temperature. Thickness of stagnant boundary layer δ, f thermal conductivity of fluid.
Boiling
(bubbles)
d
A

n ,y

q

q  T
Outer flowthermal
boundary
layer
 
f

Tf
Tf

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Example heating sphere
Temperature distribution inside a solid sphere
cp
 Ts
t

s 
r
2
r
(r
2
 Ts
r
It is correct only as soon
as the heat flux q or the
temperature is uniform
on the sphere surface
)
Boundary condition (convection)
s
 Ts
r
Heat flux calculated from
Fourier law inside the
sphere equals the flux in
fluid
  (T f  T s )
Fourier equation can be integrated at the volume of body (sphere in this case)

cp
V
 Ts
t
dv 
  s  Ts d v     q d v
2
V
V
The integrals can be evaluated by the mean value and by Gauss theorem,
assuming uniform flux at the surface
Mcp
 Ts
t
   n qdS   n qS   (T f  T s ) S
S
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Example heating sphere
For the case that the temperature inside the sphere is uniform (as soon as the
thermal conductivity s is very high) the mean temperature is identical with the
surface temperature
d Ts
dt

S
cpM
(T f  T s )
T s  T f  (T 0  T f ) e
 
 t /
cpM
S
This exponential solution works only for small values of Biot number
Bi 
D
s
 0 . 1 Thermal resistance of fluid >> thermal resistance of solid
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Convection – Nu,Re,Pr
Heat transfer coefficient  depends upon the flow velocity (u), thermodynamic
parameters of fluid () and geometry (for example diameter of sphere or pipe D).
Value  is calculated from engineering correlation using dimensionless criteria
Nu 
D
uD 
Re 
Pr 



Nusselt number (dimensionless , reciprocal thickness of boundary layer)
Reynolds number (dimensionless velocity, ratio of intertial and viscous forces)
Prandl number (property of fluid, ratio of viscosity and temperature diffusivity)
a
Rem:  is dynamic viscosity [Pa.s],  kinematic viscosity [m2/s], =/
And others
Pe=Re.Pr
Gz=Pe.D/L
De=Re√D/Dc
Péclet number
Graetz number (D-diameter, L-length of pipe)
Rayleigh
Dean number (coiled tube, Dc diameter of curvature)
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Convection in a pipe
Basic problem for heat transfer at internal flows: pipe (developed velocity
profile) and a constant wall temperature
Liquid flows in a pipe with the constant wall temperature Tw that is different than
the inlet temperature T0. Temperature profile depends upon distance from inlet and
upon radius r (only thin temperature boundary layer of fluid is heated). Heat flux
varies along the pipe even if the heat transfer coefficient  is constant, because
driving potential – temperature difference between wall and the bulk temperature
Tm depends upon the distance x. Tm is the so called mean calorific temperature
Tm ( x) 
1
 uTdA

m
A
q ( x )   ( x )(Tw  Tm ( x ))
Heat flux from wall to bulk (
is related to the calorific
temperature as a
characteristic fluid
temperature at internal flows)
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Convection in a pipe
Axial temperature profile Tm(x) follows from the enthalpy balance of system,
consisting of a short element of pipe dx :
0  m ( h ( x )  h ( x  dx ))   (Tw  Tm ( x )) D dx
0  m c p (T m ( x )  T m ( x  dx ))   (T w  T m ( x )) D dx
Tw
T0
m c p
Q
Tm
dT m
Tw  Tm
  D  dx
D
Solution Tm(x) by integration
x
dx
m c p ln
Tw  Tm ( x )
T w  T0
  D x
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Convection in a pipe
Previous integration is correct only if  and the wall temperature are constant.
 This doesn’t hold in laminar flow characterized by gradual development of thermal
boundary layer (at entry this layer is thin and therefore =/ is high,  decreases with
increasing distance). Typical correlations for laminar flow
Leveque N u 
D

 1 .6 3 R e  P r 
D
Haussen
Nu 
L
 ln D

 3.66 
0.0668 G z
1  0.04 G z
  is almost constant at turbulent flows characterized by fast development of
thermal boundary layer. Typical correlation (Dittus Boelter)
D
0.8
Nu 

 0.023 R e
Pr
2/3
1/ 3
 More complicated are cases with mixed convection (effect of temperature
dependent density and gravity), variable viscosity and first of all influence of phase
changes (boiling/condensation).
Tw
general formula for variable wall
temperature and variable heat
transfer coefficient
Q
Toutlet
T0
D
Toutlet
mcp

T0
L
dTm
T w  Tm
L
  D   dx
0
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Convection Laminar Leveque
Leveque method is very important technique how to estimate thickness of
thermal boundary layer and the heat transfer coefficient  in many internal
flows (not only in circular pipes). This theory is applicable only for “short”
channels, in the region of developing temperature profile.
Tw

T0
y
D
umax
velocity at bouindary layer
x
u x ( ) 
time of penetration
t
x
u x ( )

2

8 u
D
Nu 
 ln D

xD
8 u
  at
 
3
 axD
8u
Graetz number
Gz=Re.Pr.D/L
 1.6 3 G z
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Mixed convection, Sieder Tate
Temperature dependendent properties of fluid are respected by correction
coefficients applied to a basic formula (Leveque, Hausen, …similar corrections
are applied in correlations for turbulent regime)
Temperature dependent viscosity results in changes of velocity profiles. In
case of heating the wall temperature is greater than the bulk temperature, and viscosity of liquid at wall
lowers. Velocity gradient at wall increases thus increasing heat transfer (look at the derivation of
Leveque formula modified for nonnewtonian velocity profiles). Reversaly, in case of cooling (greater
viscosity at wall) heat transfer coefficient is reduced. This effect is usually modeled by
Sieder Tate correction (ratio of viscosities at bulk and wall temperature).
Temperature dependent density combined with acceleration (gravity)
generate buoyancy driven secondary flows. Resulting effect depends upon orientation
(vertical or horizontal pipes should be distinguished). Intensity of natural convection (buoyancy) is
characterized by Grashoff number Gr)
Mixed convection
(Grashoff)
Levequ
e
N u  1, 618 G z
1/ 3
Sieder Tate
correction
1  0, 0111G z 1/ 3 G r 4 / 9 


Gr 
g  TD

2
3
,
1/ 3
  



 W 
0 ,14
,
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Convection Turbulent flow
Boccioni
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Convection Turbulent flow
Turbulent flow is characterised by the energy transport by turbulent eddies which is
more intensive than the molecular transport in laminar flows. Heat transfer
coefficient and the Nusselt number is greater in turbulent flows. Basic differences
between laminar and turbulent flows are:
Nu is proportional to
3
0.8
in turbulent flow.
u in laminar flow, and u
Nu doesn’t depend upon the length of pipe in turbulent flows significantly (unlike
the case of laminar flows characterized by rapid decrease of Nu with the length L)
Nu doesn’t depend upon the shape of cross section in the turbulent flow regime (it
is possible to use the same correlations for eliptical, rectangular…cross sections
using the concept of equivalent diameter – this cannot be done in laminar flows)
The simplest correlation for hydraulically smooth pipe designed by Dittus Boelter
is frequently used (and should be memorized)
N u  0.023 R e
0.8
Pr
m
m=0.4 for heating
m=0.3 for cooling
Similar result follows from the Colburn analogy
Nu  0 . 023 Re
0 .8
Pr
1/3
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Pressure drop, friction factor
Pressure drop is calculated from
Darcy Weissbach equation
Friction factor f
depends upon Re
and relative
roughness
p 
1
2
 f u
2
L
D
HP4
Turbulent boundary layer
Rougness of wall has an effect upon the pressure drop and heat transfer only
if the height of irregularities e (roughness) enters into the so called buffer
layer of turbulent flow. Smaller roughness hidden inside the laminar (viscous)
sublayer has no effect and the pipe can be considered as a perfectly smooth.
y
Buffer
layer
Laminar
sublayer
Dimensionless
distance from wall
Velocity profile
Friction
velocity
e-roughness
u y
*

y 

Thickness of laminar sublayer is at value y+=5
,
u 
*
w

,
w 
dp D
dx 4
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Example smooth pipe
Calculate maximum roughness at which the pipe D=0.1 m can be
considered as smooth at flow velocity of water u=1 m/s.
Re 
p
L

uD 

1
2D
f 
 10000
 f  u  1600
e 5  10 
2
L
D p
0.316
4
 0.0316
Re
Blasius correlation for
friction factor (smooth
pipes)
 25  m
Thickness of laminar
sublayer (y+=5)
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EXAM
HP4
Heat transfer (Fourier Kirchhoff transport equation
explained in more details in the course Momentum Heat and Mass transfer)
cp (
T
t
 u T ) 
convection
Dp
Dt
this term is zero for
incom pressible liquid
  q  : 
conduction
2
(  T )
viscous
heating
Q
(R)
reaction heat
phase change enthal py
electric heat
HP4
What is important (at least for exam)
Dimensionless criteria
D
Nusselt
Nu 
Biot
Bi 
Fourier
Fo 
Reynolds
Re 
Prandtl
Pr 
Peclet
Pe  Re .Pr
Graetz
G z  R e . P r .D / L
Rayleigh
Ra 

 flu id
D
 th erm a l
D
thermal resistance in solid / thermal resistance in fluid
 solid
at
D
dimensionless time related to the penetration time through distance D
2
uD 
ratio of inertial and viscous forces


a

reciprocal thermal boundary layer

a
 g  TH
a
cp


3
ratio of momentum and temperature diffusivities
HP4
What is important (at least for exam)
Conduction - temperature field in solids
Steady heat transfer
 T
2
0 
x
2
 T
2

y
2
 T
2

z
0
2
1 
r r
(r
T
r
0
)
1 
r r
2
cylindrical c.s.
Thermal resistance RT
(r
2
T
r
)
spherical c.s.
R2
Q 
T1  T 2
R1
T1 T2
RT
RT 
ln R 2 / R1
2  L
L
Unsteady heat transfer (wave of thermal disturbance)
T
 T
2
 a(
 T
2

 T
2

)
t
x
y
z
Penetration depth (distance travelled by temperature disturbance
in time t)
2
 
2
4at
2
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What is important (at least for exam)
Convection – heat transfer from fluid to solid (-heat transf.coef.)
q   (T f  T w ) 
f

(T f  T w )
Forced heat transfer in a pipe
Laminar flow (Leveque)
Turbulent flow (Dittus Boelter)
Nu 
Nu 
D

D

 1 .6 R e  P r 
3
D
L
 0.023 R e
0.8
Pr
Pressure drop in pipes, effect of roughness and Moody diagram
p 
1
2
 f u
2
L
D
1/ 3