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HEAT PROCESSES

HP4

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 natural convection).

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

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)

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

Heat flux and conduction

dA

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 relationshipsbetween 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

Fourier’s law of heat conduction

Heat flow through boundary is projection of heat flux to the outer normal

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:

Thermal conductivity of nonmetals and gases increases with temperature (by about 10% at heating by 100K), at liquids and metals usually decreases.

Conduction – Fourier equation

dA

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

Conduction – Fourier equation

important!

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.

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 ofT(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)

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)

Conduction – thermal resistance

Q

T2

T1

T1 T2

T2

S2

R2

S

h

R1

T1 T2

S1

R1

T1

L

L

h

h1 h2

HP4

Knowing temperature field and thermal conductivity it is possible to calculate heat fluxes and total thermal power Qtransferred between two surfaces with different (but constant) temperatures T1 a T2

RT[K/W] thermalresistance

In this way it is possible to express thermal resistance of windows, walls, heat transfer surfaces …

SerialParallel Tube wall Pipe burried under surface

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=/cpis the ratio of temperature conductivity and thermal inertia

Example Conduction - nonstationary

HP4

- Typical application:How long must be a canheld 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 t=0
- 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 Fourier’s method of separated variables, see next slide

=at, where a is temperature diffusivity of meat in a can

Initial temperature

wall temperature

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=at).

Resulting equations can be solved again by the separation of variables

X and R satisfy zero boundary condition (X=R=0 at surface) and initial condition (X=R=1)

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

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 (X=R=1) 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

Example Conduction - nonstationary

HP4

I wrote previous pages at home when I have only the Baehr’s 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)

Conduction - nonstacionary

Tw

t

x

T0

δ

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 x=L 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 x=0 (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)

Conduction - nonstacionary

Tw()

T(t,x)

t

x

d

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

Theory of penetration depth

Tw

important!

t+t

t

x

T0

δ

+Δ

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)

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

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 <<R)
- For very long times the temperature profile can be approximated only by the first term of Fourier’s expansion, in case of the can by

R

L

Convection

Boiling (bubbles)

,y

dA

Outer flow-thermal boundary layer

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), Twis wall temperature. Thickness of stagnant boundary layerδ, f thermal conductivity of fluid.

Tf

Tf

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

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

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], =/

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, Dcdiameter of curvature)

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 Twthat 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 Tmdepends upon the distance x. Tmis 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)

Convection in a pipe

Q

Tw

T0

Tm

D

x

dx

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

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 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

Convection Laminar Leveque

Tw

T0

y

D

umax

x

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.

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 Gz=Re.Pr.D/x

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

Remark: further on the index ln will be frequently omitted, and Nu, denotes mean values along the whole channel.

Graetz number Gz=Re.Pr.D/L

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.

Convection Laminar Hausen

HP4

- Léveque solution is valid only at
- Laminar flow (Re<2300) and fully developed velocity profile
- In the region of thermal boundary layer development (“short” pipes, Gz>50)

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.

Hausen’s 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

Example Graetz number=50

HP4

Hydraulic stabilization length

Thermal stabilization length

Maximum length for Leveque Stabilization velocity for laminar/turbulent flow

Water: kinematic viscosity 10-6 m2/s, Pr=6

Rohsenow chapter 5.26

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

Convection Turbulent flow

important!

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

Similar result follows from the Colburn analogy

Convection Turbulent flow

HP4

- Dittus Boelter correlation assumes that Nu is independent of L. For very short pipes (L/D<60) Hausen’s correlation can be applied

2300<Re<105 0,6<Pr<500

- 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 Pětuchov (1970) recommended in VDI Warmeatlas

Friction factor

n=0,11 TW>T (heating)

n=0,25 TWT (cooling)

104Re5,1060,5Pr20000,08W /40.

Pressure drop, friction factor

important!

HP4

Pressure drop is calculated from Darcy Weissbach equation

Friction factorfdependsupon Re and relativeroughness

Turbulent boundary layer

Velocity profile

Buffer layer

Laminar sublayer

e-roughness

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 y+=5

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)

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