HEAT PROCESSES

1
HEAT PROCESSES
HP4
Heat transfer
Mechanisms of heat transfer. Conduction,
convection (heat transfer coefficients),
radiation (example cooling cabinet). Fouriers
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, Moodys diagram.
Effects of variable properties (Sieder Tate
correction for temperature dependent viscosity,
mixed and natural convection).
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
2
Mechanisms of heat transfer
HP4

• 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)

3
Heat flux and conduction
HP4
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.
Benton
4
Heat flux and conduction
HP4
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.
Intenzity of heat transfer through element dA of
boundary is characterized by vector of heat flux
W/m2
Direction and magnitude of heat flux is
determined by gradient of temperature and thermal
conductivity of media ?
Fouriers law of heat conduction
Heat flow through boundary is projection of heat
flux to the outer normal
5
Thermal conductivity ?
HP4
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 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.
6
Conduction Fourier equation
HP4
Distribution of temperatures and heat fluxes in a
solid can be expressed in differential form,
based upon enthalpy balancing of infinitesimal
volume dv


Integrating this differential equation in a
finite volume V the integral enthalpy balance can
be expressed in the following form using Gauss
theorem
Accumulation of enthalpy in unit control volume
Divergence of heat fluxes (positive if heat flows
out from the control volume at the point x,y,z)
Heat transferred through the whole surface S
Accumulation of enthalpy in volume V
7
Conduction Fourier equation
HP4
Heat flux q as well as the enthalpy h can be
expressed in terms of temperatures, giving
partial differential equation Fourier equation



Internal heat source (e.g. enthalpy change of a
chemical reaction or a volumetric heat produced
by passing electric current or absorbed
microwaves)
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.
8
Conduction - stationary
HP4
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) Boundary conditions at each
point of surface must be prescribed either
temperature T or the heat flux (for example q0
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.
The same equation written in cylindrical and
spherical coordinate system (assuming axial
symmetry)
9
Example temperature profile in a cylinder
HP4
Calculate radial temperature profile in a
cylinder and sphere (fixed temperatures T1 T2 at
inner and outer surface)
R1
R2
cylinder
Sphere (bubble)
10
Conduction thermal resistance
HP4
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
RT K/W thermal resistance
In this way it is possible to express thermal
resistance of windows, walls, heat transfer
surfaces
Serial Parallel
Tube wall Pipe burried under surface
11
Conduction - nonstacionary
HP4
Time development of temperature field T(t,x,y,z)
in a homogeneous solid body without internal heat
sources is described by Fourier equation with
the boundary conditions of the same kind as in
the steady state case and with initial conditions
(temperature distribution at time t0). This
solution T(t,x,y,z) can be expressed for simple
geometries in an analytical form (heating brick,
plate, cylinder, sphere) or numerically.
The coefficient of temperature diffusivity
a?/?cp is the ratio of temperature conductivity
and thermal inertia
12
Example Conduction - nonstationary
HP4

• Typical application How long must be a can held
  in a sterilizer so that all microorganisms at the
  can center will be killed?
• Simple answer for infinitely long time. This is
  never possible to kill all of them, only some
  prescribed percents, e.g. the number of
  Clostridium Botulinum spores should be reduced by
  12 orders according to regulation. Therefore only
  one of 1012 spores could survive the thermal
  treatment (not the viable form it would be much
  easier). Number of survived pathogens depends
  upon the whole temperature history at a given
  place (in this case in the center of can) and can
  be evaluated as an integral of temperature (see
  lecture aseptic processes).
• Temperature field in a can is described by
  Fourier equation rewritten by using dimensionless
  temperature ? that is zero on the surface and 1
  inside the can at time t0
• There are many ways how to solve the problem
• Numerically (finite differences seems to be an
  obvious choice)
• Using integral transforms (e.g. Laplace transform
  to transform differential equation to an
  algebraic one)
• Using Fouriers method of separated variables,
  see next slide

?at, where a is temperature diffusivity of meat
in a can
Initial temperature
wall temperature
13
Example Conduction - nonstationary
HP4
Temperature field ?(?at,x,r) can be decomposed
to the product X(?,x)R(?,r) of two functions that
fulfill the following two simplified
equations It can be easily verified that the
product ?(?,x,r) X(?,x)R(?,r) satisfies not only
the original Fourier equation but also the
prescribed boundary (?0) and initial (?1)
conditions (this is the reason why it was
necessary to introduce dimensionless temperature
and transformed time ? a t ). Resulting
equations can be solved again by the separation
of variables
X and R satisfy zero boundary condition (XR0 at
surface) and initial condition (XR1)
Axial temperature profile
Radial temperature profile
Sometimes (for very long time ?) it is sufficient
to approximate the solution only by the first
term in these series
14
Example Conduction - nonstationary
HP4
Resulting ORDINARY differential equations have
the following solutions (the same exponential
term describes time profile, the sin-function
axial variation of temperature and the Bessel
function J0 radial variation) Eigenvalues ?i
are determined from boundary conditions, e.q.
sin(?iL)0. Remark It is possible to modify this
solution for more complicated boundary conditions
with finite thermal resistance on the surface of
can, see Baehr S. Wärme und Stoff-übertragung,
Springer, Berlin, 1994. Coefficients xi,ri must
be calculated so that the initial conditions
(XR1) will be satisfied (using special
properties of eigenfunctions sin,J0, called
orthogonality). Seems too complicated? For more
details look at the mentioned book (Baehr), or
wait to the next semester, where the same problem
will be discussed in more details during the
course Numerical analysis of processes.
L
15
Example Conduction - nonstationary
HP4
I wrote previous pages at home when I have only
the Baehrs book at hand. Today (at work) I can
add reference to the famous book (first published
at 1946!) Carslaw H.S., Jaeger J.C. Conduction
of heat in solids, Clarendon Press, Oxford, 2004
(previous problem is solved on pages 328-329)
16
Conduction - nonstacionary
HP4
Previous example was probably too complicated.
However, there exists much simpler and probably
more important 1dimensional problem of heated
halfspace, and the step wise change of surface
temperature as a boundary condition.
This equation is the same as previously and
therefore the same method (separation of
variables) can be used. The analyzed case of heat
transfer in a plate was complicated by the
boundary condition at xL and now this
requirement is shifted to infinity. Therefore the
solution in the vicinity of surface will be
controlled only by the boundary condition at x0
(for short times) and a new kind of solution,
based upon similarity transformation can be used.
This transformation introduces suitable
dimensionless combination of time and coordinate
?, for example ??/x2, ?x2/?, ?x/??, recasting
the partial differential equation (PDE) to an
ordinary differential equation (ODE). The
simplest form of the resulting ODE is obtained
for ?x/?? (check other possibilities)
Erfc is complementary error function (available
also in some pocket calculators)
17
Conduction - nonstacionary
HP4
Erfc function describes temperature response to a
unit step at surface (jump from zero to a
constant value 1). The case with prescribed time
course of temperature at surface Tw(t) can be
solved by using the superposition principle and
the response can be expressed as a convolution
integral.
Temperature at a distance x is the sum of
responses to short pulses Tw(?)d?
Time course Tw(t) can be substituted by short
pulses
The function E(t,?,x)E(t-?,x) is the impulse
function (response at a distance x to a
temperature pulse of infinitely short duration
but unit area Dirac delta function). The
impulse response can be derived from derivative
of the erfc function
18
Theory of penetration depth
HP4
Still too complicated? Your pocket calculator is
not equipped with the erf-function? Use the
acceptable approximation by linear temperature
profile
Integrate Fourier equations (up to this step it
is accurate)
Approximate temperature profile by line
Result is ODE for thickness ? as a function of
time
Using the exact temperature profile predicted by
erf-function, the penetration depth slightly
differs ??(?at)
19
Theory of penetration depth
HP4
???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 Cp386
J/kg/K
20
Example Continuation the can
HP4

• Summary of previous results
• For very short times and small penetration depth
  the temperature profile can be always
  approximated by erf-function or linear function
  (even for cylinder as soon as the thickness ?ltltR)
• For very long times the temperature profile can
  be approximated only by the first term of
  Fouriers expansion, in case of the can by

R
L
21
Convection
HP4
General form of transport Fourier Kirchhoff
equation
Benton
22
Convection
HP4
Calculation of heat flux q from flowing fluid to
a solid surface requires calculation of
temperature profile in the vicinity of surface
(see previous Fourier Kirchhoff equation but also
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))
Tf is temperature of fluid far from surface
(behind the boundary of thermal boundary layer),
Tw is wall temperature. Thickness of stagnant
boundary layer d, ?f thermal conductivity of
fluid.
Tf
Tf
?
23
Example heating sphere
HP4
It is correct only as soon as the heat flux q or
the temperature is uniform on the sphere surface
Temperature distribution inside a solid sphere
Boundary condition (convection)
Heat flux calculated from Fourier law inside the
sphere equals the flux in fluid
Fourier equation can be integrated at the volume
of body (sphere in this case)
The integrals can be evaluated by the mean value
and by Gauss theorem, assuming uniform flux at
the surface
24
Example heating sphere
HP4
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
This exponential solution works only for small
values of Biot number Thermal resistance of
fluid gtgt thermal resistance of solid
25
Convection Nu,Re,Pr
HP4
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 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)
Rem ? is dynamic viscosity Pa.s, ? kinematic
viscosity m2/s, ??/?
And others PeRe.Pr Péclet number GzPe.D/L Gra
etz number (D-diameter, L-length of
pipe) Rayleigh DeRevD/Dc Dean number (coiled
tube, Dc diameter of curvature)
26
Convection in a pipe
HP4
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
Heat flux from wall to bulk (? is related to the
calorific temperature as a characteristic fluid
temperature at internal flows)
27
Convection in a pipe
HP4
Axial temperature profile Tm(x) follows from the
enthalpy balance of system, consisting of a short
element of pipe dx
Solution Tm(x) by integration
28
Convection in a pipe
HP4

• Previous integration is correct only if ? and the
  wall temperature are constant.
• This doesnt 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 correlation for laminar flow is Leveque
  formula
• ? is almost constant at turbulent flows
  characterized by fast development of thermal
  boundary layer. Typical correlation (Dittus
  Boelter)
• 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).
  

general formula for variable wall temperature
and variable heat transfer coefficient
Q
Tw
T0
D
Toutlet
L
29
Convection Laminar Leveque
HP4
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.
30
Convection Laminar Leveque
HP4
Linear approximation of velocity profile in
thermal boundary layer
Linear approximation of temperature profile in
thermal boundary layer
Linearized velocity profile
Tw
?
T0
y
D
umax
Transit time of particle at the distance ? from
wall
2umax
x
Thickness of boundary layer from penetration
theory
Graetz number for distance x from inlet
GzRe.Pr.D/x
31
Convection Laminar Leveque
HP4
Local Nux decreases with x. Mean value of Nu
corresponding to the length of pipe L is obtained
by integration, so that the outlet temperature
Toutlet can be calculated from enthalpy balance
Heat capacity of stream
Heat transfer surface
Mean heat transfer coefficient related to the
mean logarithmic temperature difference
Graetz number GzRe.Pr.D/L
Remark further on the index ln will be
frequently omitted, and Nu, ? denotes mean values
along the whole channel.
32
Convection Laminar Leveque
HP4
Previous result holds only for parabolic velocity
profile, corresponding to the fully developed
flow of a Newtonian liquid. For fluids with
different velocity profiles (e.g. profiles
corresponding to power law liquids) the whole
derivation must be repeated the idea of Leveque
remains
Linear approximation of power law velocity
profile in thermal boundary layer
Tw
?
T0
y
D
umax
x
Velocity profile for power law liquid (n-flow
index)
The same procedure can be applied to the case
with the constant wall temperature and with a
constant wall heat flux (only the constant c
differs)
This example demonstrates that sometimes it is
more important to know a derivation than the
final result.
33
Convection Laminar Hausen
HP4

• Léveque solution is valid only at
• Laminar flow (Relt2300) and fully developed
  velocity profile
• In the region of thermal boundary layer
  development (short pipes, Gzgt50)

For a pipe of arbitrary length the Graetz
solution in form of series of eigenfunctions (see
the Fourier method of separation of variables)
can be used. At a very large distance from inlet
the boundary layers merge, transverse temperature
profiles become similar and the heat transfer
coefficient approaches limiting value (Nu?3.66
for the case of constant wall temperature,
Nu?4.36 for constant heat flux).
Constant heat flux (therefore increasing wall
temperature). Typical for the counterflow heat
exchangers.
Hausens semiempirical correlation is a blend of
Léveque solution and asymptotic values of Nu?
Constant wall temperature (maintained by boiling
or condensing steam)
Prove that the Hausen formula reduces to Leveque
for Gz??
34
Example Graetz number50
HP4
Hydraulic stabilization length
Thermal stabilization length
Maximum length for Leveque
Stabilization velocity for laminar/turbulent flow
Water kinematic viscosity 10-6 m2/s, Pr6
Rohsenow chapter 5.26
D m U m/s Re Lh m LT m
0.1 1 105 1.2 turbulent
0.05 1 50000 0.5 turbulent
0.01 1 10000 0.07 turbulent
0.01 0.1 1000 0.5 1.2
0.01 0.01 100 0.05 0.12
0.005 0.1 500 0.12 0.3
0.005 0.01 50 0.012 0.03
35
Mixed convection, Sieder Tate
HP4

• 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)
Sieder Tate correction
Leveque
36
Convection Turbulent flow
HP4
Boccioni
37
Convection Turbulent flow
HP4

• 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 in laminar flow,
  and in turbulent flow.
• Nu doesnt 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 doesnt depend upon the shape of cross section
  in the turbulent flow regime (it is possible to
  use the same correlations for eliptical,
  rectangularcross 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)
m0.4 for heating m0.3 for cooling
Similar result follows from the Colburn analogy
38
Convection Turbulent flow
HP4

• Dittus Boelter correlation assumes that Nu is
  independent of L. For very short pipes (L/Dlt60)
  Hausens correlation can be applied

2300ltRelt105 0,6ltPrlt500

• Effect of wall roughness can be estimated from
  correlations based upon analogies between
  momentum and heat transfer (Reynolds analogy,
  Colburn analogy, Prandtl Taylor analogy). Results
  from hydraulics (pressure drop, friction factor
  ?f) are used for heat transfer prediction.
  Example is correlation Petuchov (1970)
  recommended in VDI Warmeatlas

Friction factor
n0,11 TWgtT (heating) n0,25 TW?T (cooling)
104?Re?5,106 0,5?Pr?2000 0,08??W /??40.
39
Pressure drop, friction factor
HP4
Pressure drop is calculated from Darcy Weissbach
equation
Friction factor ?f depends upon Re and relative
roughness
40
Turbulent boundary layer
HP4
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
Dimensionless distance from wall
Friction velocity
Thickness of laminar sublayer is at value y5
41
Example smooth pipe
HP4
Calculate maximum roughness at which the pipe
D0.1 m can be considered as smooth at flow
velocity of water u1 m/s.
Blasius correlation for friction factor (smooth
pipes)
Thickness of laminar sublayer (y5)