1. HEAT PROCESSES Heat transfer HP4
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 natural convection).
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010 1

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) 2

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 3

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] dA
Direction and magnitude of heat flux is determined by gradient of temperature and thermal conductivity of media 
Fourier’s law of heat conduction
Heat flow through boundary is projection of heat flux to the outer normal 4

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. 5

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) dA
Heat transferred through the whole surface S
Accumulation of enthalpy in volume V 6

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)
important!
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. 7

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 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.
The same equation written in cylindrical and spherical coordinate system (assuming axial symmetry) 8

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
cylinder R2
Sphere (bubble) 9

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
h1 h2 S T1 T2 h S1 S2
T1 T2 L R1 R2
T1 T2 Q
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 10

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 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.
The coefficient of temperature diffusivity a=/cp is the ratio of temperature conductivity and thermal inertia 11

12. Theory of penetration depth
HP4
Development of temperature profile in a half-space. Use the acceptable approximation by linear temperature profile Tw T0 x δ
t+t t
+Δ
important!
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) 12

13. 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
Cp=386 J/kg/K  13

14. Convection General form of transport Fourier Kirchhoff equation HP4
Benton 14

15. Convection
HP4
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)])
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. dA
Boiling (bubbles)
Outer flow-thermal boundary layer ,y Tf Tf  15

16. 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 16

17. 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 >> thermal resistance of solid 17

18. Convection – Nu,Re,Pr important! 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], =/
important!
And others
Pe=Re.Pr Péclet number
Gz=Pe.D/L Graetz number (D-diameter, L-length of pipe)
Rayleigh
De=Re√D/Dc Dean number (coiled tube, Dc diameter of curvature)

19. 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)

20. 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 : x dx D T0 Tw Tm Q
Solution Tm(x) by integration

21. Convection in a pipe
HP4
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
 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).
Leveque
Haussen
general formula for variable wall temperature and variable heat transfer coefficient Q Tw T0 D
Toutlet L

22. 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. x D T0 Tw  y
umax
velocity at bouindary layer
time of penetration
Graetz number Gz=Re.Pr.D/L

23. 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

24. Convection Turbulent flow
HP4
Boccioni

25. 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 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)
m=0.4 for heating
m=0.3 for cooling
important!
Similar result follows from the Colburn analogy

26. Pressure drop, friction factor
HP4
Pressure drop is calculated from Darcy Weissbach equation
important!
Friction factor f depends upon Re and relative roughness

27. 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.
Velocity profile
Laminar sublayer
e-roughness
Buffer layer y
Dimensionless distance from wall
Friction velocity
Thickness of laminar sublayer is at value y+=5

28. Example smooth pipe
HP4
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.
Blasius correlation for friction factor (smooth pipes)
Thickness of laminar sublayer (y+=5)

29. HP4

30. EXAM
HP4
Heat transfer (Fourier Kirchhoff transport equation explained in more details in the course Momentum Heat and Mass transfer)

31. What is important (at least for exam)
HP4
Dimensionless criteria
Nusselt
Biot
Fourier
Reynolds
Prandtl
Peclet
Graetz
Rayleigh
reciprocal thermal boundary layer
thermal resistance in solid / thermal resistance in fluid
dimensionless time related to the penetration time through distance D
ratio of inertial and viscous forces
ratio of momentum and temperature diffusivities

32. What is important (at least for exam)
HP4
Conduction - temperature field in solids
Steady heat transfer
Thermal resistance RT L R1 R2
T1 T2
Unsteady heat transfer (wave of thermal disturbance)
Penetration depth (distance travelled by temperature disturbance in time t)

33. What is important (at least for exam)
HP4
Convection – heat transfer from fluid to solid (-heat transf.coef.)
Forced heat transfer in a pipe
Laminar flow (Leveque)
Turbulent flow (Dittus Boelter)
Pressure drop in pipes, effect of roughness and Moody diagram