Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Heat Transport and Transfer Yannick Hallez LGC-UPS
29 f´evrier 2012
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Introduction Quantity of interest As far as heat (energy) is concerned the quantity of interest is the heat quantity per unit of volume : ρcp T in J/m3 in SI units. More generally it has the dimension ML−1 T −2 . ρ is the fluid or solid density. cp is its specific heat capacity in Jkg −1 K −1 (cp = 4185 J/kg/K for water for instance). In this course we suppose ρ and cp are constants. But we turn on an equation on the temperature Heat is the extensive quantity and it is possible to write a heat conservation principle. It is not possible on temperature, which is intensive. But in the end, we will write a transport equation for the temperature field. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Introduction
Flux density The heat flux density will be denoted q. In SI units it is expressed as Js−1 m−2 or W/m2 . It has a dimension ML−3 T −2 . It is a vector, as denoted by the bold font.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Analogy between heat and mass transfer Conservation of ... We search ... Flux density units
Mass Transfer Mass C (kg/m3 ) C (kg/m3 ) kgs−1 /m2
Heat Transport and Transfer
Heat Transfer Energy ρcp T (J/m3 ) T (K) W/m2
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Plan
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Mechanisms : an overview Heat source or sink Chemical reaction (e.g. combustion) Nuclear reaction Joule effect Radiation Viscous dissipation (space shuttle re-entry ) Mass transport mechanisms Advection Conduction/Diffusion Convection is actually an Advection/Diffusion mixing. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
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Heat transport Mechanisms Advection Conduction/Diffusion
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Heat transport equation
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Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
Advection Conduction/Diffusion
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by advection Consider a blob of hot fluid inside a cold stream with a velocity field u. At any point inside the fluid, the advective heat flux density is q = ρcp T u. It means that the hot spot is just transported from a point to another with a velocity u. The advective flux of any quantity A is just Au. In mass transport A is the temperature (mass per unit volume), and in heat transport A is the heat per unit volume.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by advection Characteristic advective time scale If the movement of fluid takes place on a length scale L and with a velocity scale U , it is possible to define a characteristic advective time scale by L (1) Ta = U
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Analogy between heat and mass transfer Conservation of ... We search ... Flux density units Advective flux density
Mass Transfer Mass C (kg/m3 ) C (kg/m3 ) kgs−1 /m2 ja = Cu
Heat Transport and Transfer
Heat Transfer Energy ρcp T (J/m3 ) T (K) W/m2 qa = ρcp T u
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
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Heat transport Mechanisms Advection Conduction/Diffusion
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Heat transport equation
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Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
Advection Conduction/Diffusion
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction Microscopic point of view Temperature is a measure of molecules/atoms agitation. The kinetic energy of molecules will be modified during interactions : the kinetic energy of a very agitated molecule will decrease when hitting a less agitated molecule. The kinetic energy of the less agitated one will rise because of the interaction. So when putting a hot zone in contact with a cold zone, the temperature tends to become homogeneous. Heat Transport and Transfer
Hot
Cold
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction/diffusion Fourier’s law The simplest macroscopic model for the diffusive flux density of heat is based on the idea that heat goes from warm zones to cold ones, so that the flux is along the temperature gradient, and in the opposite direction. So let λ be the proportionality factor, and we get q = −λ∇T (2) This is Fourier’s law, and λ is called the thermal conductivity in W/(mK) in SI units. It is also possible to define the heat diffusion coefficient of the solid/liquid D = ρcλp which has a dimension L2 T −1 (m2 /s in SI units).
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction/diffusion
Examples of thermal conductivities Material Copper Steel Glass, brick Water Glass wool Air
Thermal condutivity λ (W m−1 K −1 ) 280 35 0.6 0.4 0.03 0.02
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Analogy between heat and mass transfer Conservation of ... We search ... Flux density units Advective flux density Diffusive flux density Diffusion coefficient
Mass Transfer Mass C (kg/m3 ) C (kg/m3 ) kgs−1 /m2 ja = Cu Fick’s law jd = −D∇C D (m2 /s)
Heat Transport and Transfer
Heat Transfer Energy ρcp T (J/m3 ) T (K) W/m2 qa = ρcp T u Fourier’s law qd = −λ∇T D = λ/(ρcp ) (m2 /s)
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction/diffusion Try to draw the diffusive flux vector on these simulation images ! Red is for high temperature, blue is for low temperature.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction/diffusion Try to draw the diffusive flux vector on these simulation images ! Red is for high temperature, blue is for low temperature.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction/diffusion Try to draw the diffusive flux vector on these simulation images ! Red is for high temperature, blue is for low temperature.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction/diffusion Characteristic diffusive time scale (no change vs mass transfer) ! Since a diffusive time scale depends on the length scale L used for observation (dimension L) and of the diffusivity coefficient D (dimension L2 T −1 ), an obvious time scale can be built as Td =
L2 D
(3)
Note that we recover the idea that a diffusive length scale grows with time as Ld = (DT )1/2 (4)
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Comparison of advection and diffusion The P´eclet number (1/2) (no change vs mass transfer) ! It is possible to compare the diffusive and advective time scales using their ratio, called the P´eclet number Pe =
UL Td = Ta D
(5)
When P e 1, the advective time scale is much shorter than the diffusive time scale, so that advection is the prevailing phenomenon. When P e 1, diffusion is much faster than advection and is the dominant phenomenon.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Comparison of advection and diffusion
The P´eclet number (2/2) (no change vs mass transfer) ! Note that the P´eclet number can also be seen as the ratio of an advective to a diffusive flux : Pe =
qa TU UL = = qd DT /L D
(6)
When P e 1 the advective flux is much larger than the diffusive flux so that advection dominates, and vice-versa.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Analogy between heat and mass transfer Conservation of ... We search ... Flux density units Advective flux density Diffusive flux density Diffusion coefficient Advection/Diffusion
Mass Transfer Mass C (kg/m3 ) C (kg/m3 ) kgs−1 /m2 ja = Cu Fick’s law jd = −D∇C D (m2 /s) P´eclet P e = UDL
Heat Transport and Transfer
Heat Transfer Energy ρcp T (J/m3 ) T (K) W/m2 qa = ρcp T u Fourier’s law qd = −λ∇T D = λ/(ρcp ) (m2 /s) P´eclet P e = UDL
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Comparison of advection and diffusion (no change vs mass transfer) !
Heat Transport and Transfer
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.00.0
Ta Td
T
Because of the linear and quadratic L dependencies of Ta and Td , whatever the values of the velocity scale and the diffusivity coefficient, for very short length scales diffusion is always faster than advection, and at large length scales advection is always faster than advection. The critical length scale Lc where the transport regime is switched from diffusive to advective corresponds to equal advective and diffusive times (or fluxes), i.e. P e = 1. Then we get Lc = D/U .
0.5
1.0 L
1.5
2.0
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Advection Conduction/Diffusion
Transport by conduction/diffusion and advection Let’s keep in mind that ... In advective processes, characteristic distances vary like t. In diffusive processes, characteristic distances vary like t1/2 . The P´eclet number P e = U L/D compares the relative importance of advective and diffusive heat transport phenomena. If P e 1 advection dominates, and if P e 1 diffusion dominates. An advective flux is written q = ρcp T u, where u is the fluid velocity (vector). A diffusive flux is given by q = −λ∇T (Fourier’s law), where λ is the thermal conductivity in W/(mK) in SI units. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Plan
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Heat transport Mechanisms
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Heat transport equation
3
Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
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Heat transport Mechanisms
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Heat transport equation From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
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Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Sketch
Let us consider a control volume V of fluid or solid, delimited by a closed surface S. Anywhere on the surface it is possible to define a unit normal vector n directed arbitrarily outside the volume V . In the whole fluid or solid domain, a non-constant flux of heat q exists.
Heat Transport and Transfer
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From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
From heat conservation to transport equation
A heat balance on the volume V says that ”the accumulation of heat inside the volume is equal to the heat flux entering the volume minus the flux of heat leaving the volume (net heat flux) plus the production of heat inside the volume“.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
From heat conservation to transport equation
If a heat field H = ρcp T is defined throughout the domain, the total heat inside the volume V is Z ρcp T dV, (7) V
and its variation in time is Z ∂ ρcp T dV ∂t V
Heat Transport and Transfer
(8)
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
From heat conservation to transport equation The heat flux density at any point is q. Hence the heat flux through a small surface element with unit normal n and surface area dS is dQ = q · ndS. So the net flux through the full closed surface S is I Q = − q · ndS. (9) S
The minus sign is due to our convention of n being directed outward the surface : a positive value of Q means heat is entering the volume. Heat Transport and Transfer
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From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
From heat conservation to transport equation
Finally, if a there is a heat source or sink s (in W/m3 ) in the volume, the total heat source in the volume V is Z sdV (10) V
Heat Transport and Transfer
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From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat conservation to transport equation
Then, the classical heat balance on the volume V becomes Z I Z ∂ ρcp T dV = − q · ndS + sdV ∂t V S V or
Z V
∂ρcp T dV = − ∂t
I
Z q · ndS +
S
(11)
sdV
(12)
V
(note that the volume boundaries are held at fixed positions).
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
From heat conservation to transport equation
With the help of Green’s formulae (divergence theorem), the first surface integral on the right-hand-side can be rewritten as a volume integral : Z Z Z ∂ρcp T dV = − div(q)dV + sdV, (13) ∂t V V V where div(q) ≡ ∇ · q is the divergence of q.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
From heat conservation to transport equation
Let’s continue... Z V
∂ρcp T + div(q) − s dV = 0. ∂t
(14)
Since the heat balance presented above is valid for any volume V , the integrand is necessarily zero in the whole space, and we get the heat transport equation valid everywhere ∂ρcp T + div(q) = s ∂t
Heat Transport and Transfer
(15)
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Other forms Let’s make appear our favorite heat fluxes, namely the advective and diffusive ones. In this case q = ρcp T u − λ∇T
(16)
and we often encounter the equation under the form of an ”advectiondiffusion equation“ ∂ρcp T + div(ρcp T u) = div(λ∇T ) + s (17) ∂t Moreover if the fluid is incompressible (∇·u = 0, see hydrodynamics lessons) and if the thermal conductivity is constant we get ∂ρcp T + u · ∇(ρcp T ) = λ∆T + s ∂t where ∆ ≡ ∇2 is the laplacian. Heat Transport and Transfer
(18)
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From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Other forms
If we suppose that ρ and cp are constant and there is no source term and we get λ ∂T + u · ∇(T ) = ∆T (19) ∂t ρcp or ∂T + u · ∇(T ) = D∆T (20) ∂t This is the transport equation for the temperature field. See the similarity with the transport equation for the concentration field !
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Analogy between heat and mass transfer Conservation of ... We search ... Flux density units Advective flux density Diffusive flux density Diffusion coefficient Advection/Diffusion Transport equation
∂C ∂t
Mass Transfer Mass C (kg/m3 ) C (kg/m3 ) kgs−1 /m2 ja = Cu Fick’s law jd = −D∇C D (m2 /s) P´eclet P e = UDL + u∇C = D∆C + r
Heat Transport and Transfer
Heat Transfer Energy ρcp T (J/m3 ) T (K) W/m2 qa = ρcp T u Fourier’s law qd = −λ∇T D = λ/(ρcp ) (m2 /s) P´eclet P e = UDL ∂T 0 ∂t + u∇T = D∆T + s
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
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Heat transport Mechanisms
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Heat transport equation From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
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Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The heat transport equation In this course we will most often use this last form ∂T + u · ∇T = D∆T (21) ∂t The second term on the left hand side is the ”advective term“, the first on the right hand side is the ”diffusive term“. This equation can be seen as a constraint between the velocity and temperature fields that ensures that the first principle of heat conservation is respected. In a general flow, the velocity could depend on T through convection effects linked to density differences. In the present form, the density is supposed to be constant and free convection effects are neglected. Here the temperature does not influence the flow, it behaves as a so-called ”passive scalar“. In this case the equation is linear. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The heat transport equation
Boundary conditions This PDE must be complemented by initial and boundary conditions. Some of them are Constant temperature on a surface : T = Ti Constant flux through a surface : ∂T /∂n = constant (where n is a coordinate normal to the surface) Symmetry (or zero flux) : ∂T /∂n = 0
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The heat transport equation Some special cases of the transport equation are : The advection equation ∂T + u · ∇T = 0 ∂t
(22)
The diffusion equation (used for example in solids) ∂T = D∆T ∂t
(23)
stationary equations ... Now, let’s try to solve the heat transport equation in some simple cases ! Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
1
Heat transport Mechanisms
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Heat transport equation From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
3
Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The concept of resistance to heat transfer Using an analogy with electricity and Ohm’s law, it is possible to define a resistance to heat transfer for steady state problems. The difference of temperature through a medium is equivalent to a difference of electrical potential and the heat flux is equivalent to the current. So e T1 − T2 and R = (24) Q= R λA where e is the medium thickness, and A is homogeneous to a surface area. Geometry thickness e definition of A plane xo − xi A=S Ao −Ai cylindrical ro − ri A = ln(Ao /Ai ) , Ao = 2πro L, Ai = 2πri L √ spherical ro − ri A = Ao Ai , Ao = 4πro2 , Ai = 4πri2 Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
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Heat transport Mechanisms
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Heat transport equation From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
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Heat transfer at interfaces
4
Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
See mass transport equation solutions and replace C by T ! ! !
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
1
Heat transport Mechanisms
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Heat transport equation From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
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Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Introduction of non-dimensional variables Let us denote the characteristic time scale of a mass transfer phenomena T , its length scale L, its temperature scale Θ, its velocity scale U . We now introduce the following non-dimensional variables : t∗ = t/T , x∗ = x/L, u∗ = u/U and T ∗ = T /Θ By construction, all the variables with a star are of order one in our problem. Do not forget that the ∇ operator is based on one space derivative and is thus of order to 1/L. Similarly, the ∆ operator is based on two successive derivatives in space, and is thus of order 1/L2 . We then introduce the scaled operators : ∇∗ = ∇/(1/L) and ∆∗ = ∆/(1/L2 ) Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
Introduction of non-dimensional variables The heat transport equation is recalled here ∂T + u∇T = D∆T ∂t We can express all the variables and operators in this equation as functions of the non-dimensional ones : t = t∗ T , x = x∗ L, u = u∗ U and T = T ∗ Θ and for the operators ∇ = ∇∗ /L and ∆ = ∆∗ /L2 Then inject everything in the transport equation : UΘ Θ ∂T ∗ Θ + u∗ ∇ ∗ T ∗ = D∆∗ T ∗ 2 ∂t∗ T L L Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The non-dimensional heat transport equation
Multiply both sides by L/(U Θ) to obtain L ∂T ∗ D ∗ ∗ ∗ +u ∇ T = ∆∗ T ∗ U T ∂t∗ LU or
1 ∂T ∗ 1 ∗ ∗ ∆ T + u∗ ∇∗ T ∗ = St ∂t∗ Pe We recognize the aforementioned P´eclet and Strouhal numbers.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The non-dimensional form : comparing different effects If T not linked to something external, it is either an advective time scale and St = 1, or a diffusive time scale and St = P e (check all this !). Let’s recall the non-dimensional heat transport equation : 1 ∂T ∗ 1 ∗ ∗ + u∗ ∇∗ T ∗ = ∆ T St ∂t∗ Pe By construction, every term involving only variables or derivatives with star superscripts is of order one. On the other hand, nondimensional numbers have values fixed by the characteristic scales of the problem (and are generally not of order one). Therefore, the above non-dimensional form provides an effective and easy way to evaluate the relative weight of the different terms in the solution. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The non-dimensional form : comparing different effects
Example Consider heat transfer around 10−5 m particles in a flow with velocity of 10 m/s in water (D = 1.4 10−7 m2 /s). There is no reaction and no external forcing. The general form of the mass transport equation in this case is ∂T + u∇T = D∆T ∂t With this problem, both diffusion and advection exist. But should be consider both or may we neglect one of the two phenomena ?
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The non-dimensional form : comparing different effects The heat transport is governed by the following non-dimensional equation : ∂T ∗ 1 ∗ ∗ ∆ T + u∗ ∇ ∗ T ∗ = ∂t∗ Pe The P´eclet number is here P e = U L/D = 10.10−5 /1.4 10−7 = 714 Thus we see that the temporal and advection terms are of order one (only “star” variables) and the diffusion term ifs of order 1/P e = 1.4 10−3 . Then we conclude that the diffusion term is negligibly small compared to the two other ones and the problem is governed by the equation ∂T ∗ + u∗ ∇∗ T ∗ = 0 ∂t∗ Yes, it is possible to neglect thermal diffusion. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
From heat balance to transport equation About the heat transport equation Steady state solutions Unsteady diffusion equation solutions Non-dimensional form of the transport equation
The non-dimensional form : a proof of the allmightiness of non-dimensional numbers Consider a heat transport problem with velocity scale U = 0.1 m/s, length scale L = 10 cm and diffusivity D = 10−5 m2 /s (something happening at large scale in a gas). The P´eclet number is P e = 10. Consider a second mass transport problem with scale U = 1 cm/s, length scale L = 1 µm and diffusivity D = 10−9 m2 /s (something happening at very small scale in a water). The P´eclet number is P e = 10. Do you believe the solution is exactly the same be cause these two problems have the same P´eclet number ? They both obey to exactly the same equation ∂T ∗ 1 + u∗ ∇ ∗ T ∗ = ∆ ∗ T ∗ , ∗ ∂t 10 so yes, the (non-dimensional) solution is the same ! Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Plan
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Heat transport Mechanisms
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Heat transport equation
3
Heat transfer at interfaces
4
Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
4
Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Thermal boundary layer Consider an interface between two phases (liquid/gaz, solid/liquid, solid/gaz) on which the temperature Ti is constant. The temperature far from the interface in the fluid is T∞ . If there is a flow along the interface and a diffusion process in the fluid between the interface and the bulk of the fluid, a so-called boundary layer appears near the interface. It correspond to a thin fluid zone in which the variation of temperature is localized. The extent of this zone δT is called the thermal boundary layer thickness. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Thermal boundary layer The film model Sometimes, if the transport through the boundary layer is mainly diffusive (low P e), the assumption that the temperature profile is linear in the boundary layer is done. It is called the film model (or film theory). In this case, the diffusive flux through the boundary layer is exactly qd = −λ
T∞ − Ti δT
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Heat transfer coefficient Heat transfer across the boundary layer Since at the interface the fluid velocity vanishes (adherence), sufficiently close to it advection becomes dominated by diffusion and the heat flux density is q = −λ∇T . In particular, it is true right at the interface, where the heat flux density through the wall is ∂T , qw = −λ ∂x w where x is the direction normal to the wall.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Heat transfer coefficient Heat transfer across the boundary layer The derivative in this expression is the slope of the temperature profile at the wall. Hence, knowledge of the temperature profile through the boundary layer is required to compute the heat transfer between the two phases. To obtain this profile, it is necessary to solve the full heat transport equation and hydrodynamics equations. This is rarely possible analytically, and often very complicated numerically.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Heat transfer coefficient Heat transfer across the boundary layer Instead, we an use an ersatz : since the heat flux is driven by a temperature variation, we write it as q = h(Ti − T∞ ), where h is the heat transfer coefficient. Note that it could also be written q = (Ti − T∞ )/R, where R is a resistance to heat transfer. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Heat transfer coefficient The heat transfer coefficient h is expressed in W/m2 /K in SI units. Note that if the boundary layer is dominated by diffusion (P e is low), the film model can be used and the heat flux density is q = h(Ti − T∞ ) = qd = −λ which shows that h=
Ti − T∞ T∞ − Ti =λ δT − 0 δT
λ δT
Hence the smaller the boundary layer thickness, the stronger the transfers (and vice-versa). (exactly the same as mass transfer)
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Analogy between heat and mass transfer Conservation of ... We search ... Flux density units Advective flux density Diffusive flux density Diffusion coefficient Advection/Diffusion Transport equation Convective flux density
Mass Transfer Mass C (kg/m3 ) C (kg/m3 ) kgs−1 /m2 ja = Cu Fick’s law jd = −D∇C D (m2 /s) P´eclet P e = UDL ∂C ∂t + u∇C = D∆C + r jc = k(Ci − C∞ ) k : mass transfer coeff.
Heat Transport and Transfer
Heat Transfer Energy ρcp T (J/m3 ) T (K) W/m2 qa = ρcp T u Fourier’s law qd = −λ∇T D = λ/(ρcp ) (m2 /s) P´eclet P e = UDL ∂T 0 ∂t + u∇T = D∆T + s qc = h(Ti − T∞ ) h : heat transfer coeff.
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Form of correlations for the heat transfer coefficient For practical problems, semi-empirical correlations can be found in the litterature for h, or rather its non-dimensional form hL/λ = N u called the Nusselt number. To guess what these correlations look like, let us use the Buckingham Π theorem. The transfer through the boundary layer depends on the flow and on the fluid/solute properties. The flow depends on the velocity scale U , a length scale L and the kinematic viscosity ν. The properties of the fluid are the fluid viscosity ν and the thermal diffusivity D. So our problem is then to find a correlation of the form h = f (U, L, ν, D)
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Form of correlations for the heat transfer coefficient We have a relation between 7 variables (h, U , L, ν, ρ, cp and λ) and all the dimensions of these variables can be expressed with units of length L, time T , mass M and temperature Θ (4 dimensions). Hence the Buckingham Π theorem says that the correlation can be written with only 7 − 4 = 3 non-dimensional numbers. To find them, let’s choose our four independent variables. For example L, ν, ρ and cp . ([L] = L, [ν] = L2 T −1 , [ρ] = ML−3 and [cp ] = L2 T −2 Θ−1 . (L is used to have lengths units, then we add ν to introduce time units, then ρ to introduce mass units and finally cp for temperature units). Thus they are independent variables.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Form of correlations for the heat transfer coefficient Hence the three remaining variables (h, U and λ) can be nondimensionalized by something of the form Lα ν β ργ cδp . This process will lead to three non-dimensional variables (or numbers). If we try to non-dimensionalize h, we must verify [h] = [Lα ν β ργ cδp ] or more precisely MT −3 Θ−1 = Lα (L2 T −1 )β (ML−3 )γ (L2 T −2 Θ−1 )δ For the power to be the same on both sides, we have to solve the system : for mass 1 = γ, for length 0 = α + 2β − 3γ + 2δ, for temperature −1 = −δ and for time −3 = −β − 2δ. Solution is (α, β, γ, δ) = (−1, 1, 1, 1) and so [h] = [νρcp /L]. Thus the first non-dimensional number is hL h = Π1 = νρcp /L νρcp Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Form of correlations for the heat transfer coefficient Now we apply the procedure to U : we must ensure [U ] = [Lα ν β ργ cδp ] (with new α, β, γ and δ exponents of course !). [U ] = LT −1 = Lα (L2 T −1 )β (ML−3 )γ (L2 T −2 Θ−1 )δ The solution is (α, β, γ, δ) = (−1, 1, 0, 0), hence [U ] = [ν/L] and the non-dimensional number is Π2 =
UL U = = Re, ν/L ν
where Re is the Reynolds number.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Form of correlations for the heat transfer coefficient Finally, we apply the procedure to λ. We must have [λ] = [Lα ν β ργ cδp ], or [λ] = MLT −3 Θ−1 = Lα (L2 T −1 )β The solution to have the same powers on both sides is (α, β, γ, δ) = (0, 1, 1, 1) so that [λ] = [νρcp ] and the last non-dimensional number is D 1 λ = = , Π3 = νρcp ν Pr where P r = ν/D is the Prandtl number. The Prandtl number is for heat transfer what the Schmidt number is for mass transfer.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Form of correlations for the heat transfer coefficient The Buckingham Π theorem states that the dimensional relation h = f (U, L, ν, λ, ρ, cp ) can be recast in the non-dimensional form Π3 = f (Π1 , Π2 ) or here hL −1 , Re Pr = f νρcp Since D = λ/(ρcp ), Π1 = relation an also be written
hL νρcp
=
hLD νλ
= N u/P r. So the above
P r−1 = f (Re, N u/P r) or equivalently N u = f (Re, P r)
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
The Nusselt number From the previous slides, it is obvious that the Nusselt number is the non-dimensional version of the heat transfer coefficient. But it is also more than that. It is actually a measure of the ratio of a convective heat transfer (heat transfer through a boundary layer caused by a non-zero fluid velocity, i.e. both advective and diffusive) to a purely diffusive heat transfer (through the boundary layer if advection was ignored). We have seen that the flux density of the convective heat transfer is qc = h(Ti − T∞ ) and we know that a diffusive flux density is qd = −λ∇T ∼ λ(Ti − T∞ )/L where L is the length scale on which temperature gradients spread or the length along the plate for a boundary layer on a flat plate. Then h(Ti − T∞ ) hL qc = = = Nu qd λ(Ti − T∞ )/L λ Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Analogy between heat and mass transfer Conservation of ... We search ... Flux density units Advective flux density Diffusive flux density Diffusion coefficient Advection/Diffusion Transport equation Convective flux density
Fluid properties
Mass Transfer Mass C (kg/m3 ) C (kg/m3 ) kgs−1 /m2 ja = Cu Fick’s law jd = −D∇C D (m2 /s) P´eclet P e = UDL ∂C ∂t + u∇C = D∆C + r jc = k(Ci − C∞ ) k : mass transfer coeff. Sherwood Sh = kL D ν Schmidt Sc = D
Heat Transport and Transfer
Heat Transfer Energy ρcp T (J/m3 ) T (K) W/m2 qa = ρcp T u Fourier’s law qd = −λ∇T D = λ/(ρcp ) (m2 /s) P´eclet P e = UDL ∂T 0 ∂t + u∇T = D∆T + s qc = h(Ti − T∞ ) h : heat transfer coeff. Nusselt N u = hL λ ν Prandtl P r = D 72 / 85
Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Thermal boundary layer Introduction of the heat transfer coefficient The Nusselt number Examples of heat transfer coefficient correlations
Examples of heat transfer coefficient correlations
See mass transfer course ! They are the same ! ...But replace Sc by P r, and replace Sh by N u.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Plan
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces
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Exercises
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
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Heat transport Mechanisms
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Heat transport equation
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Exercises Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Double glazing A double glazing is made with two 2 m × 2 m glass plates whose thickness is e = 3 mm. These plates are separated by a gap h = 8 mm. Ambient temperature is Ti = 20˚C in the room, and To = 0˚C outside. Compute the heat flux (loss) in Watts across the double glazing. Compute the temperature at interfaces and draw the temperature profile. Compute the size of the boundary layers (approximately). Re-compute previous results assuming the glass plates are touching. What would be the thickness e0 of a unique glass plate insulating as well as the double glazing ? Data : λair = 0.02 W/m/K, λglass = 0.06 W/m/K, ho = 8 W/m2 /K, hi = 5 W/m2 /K. Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces
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Exercises Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Insulation of a steam duct
A steam duct has an 80 mm diameter. It is coated with refractory cement (λ = 0.48 W/m/K) to reduce heat loss. The heat transfer coefficient at the cement surface is 8 W/m2 /K. Compute the heat flux Q (loss) for a tube length L as a function of the cement thickness e. Compute the worst thickness ec for insulation (called critical thickness) Compute the minimum thickness necessary to improve the insulation over a naked duct.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Insulation of a steam duct
Q
0.35 0.34 0.33 0.32 0.31 0.30 0.29 With insulation 0.28 Without insulation 0.27 0.00 0.02 0.04 0.06 0.08 0.10 0.12 e Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
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Heat transport Mechanisms
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Heat transport equation
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Heat transfer at interfaces
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Exercises Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Sticking plates on the Moon We want to stick two 1 m2 plates of plastic and cork. The thickness of the platic plate is ep = 12.5 cm. The thickness of the cork plate is ec = 25 cm. Plates are stuck using a thin glue film that must be maintained at 43˚C for a long time period. To achieve this, the external part of the plastic plate is heated at a temperature Tp . On the other side, the external face of the cork plate is facing a room at ambient temperature Ta = 21˚C. Free convection takes place between the external cork face at temperature Ts and the ambient air of the room at Ta . The heat transfer coefficient is h = 11.2 W/m2 /K. Convective effects on the external plastic plate at temperature Tp are neglected. Data : λplastic = 2.24 W/m/K, λcork = 0.29 W/m/K.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Sticking plates on the Moon
Find the resistances to heat transfer Rp in the plastic, Rc in the cork and Rconv between the cork plate and the room. Compute the heat flux through the system. Compute the temperature Tp that must be maintained so that the glue layer is at 43˚C.
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Sticking plates on the Moon In free convection on a vertical plate, the Nusselt number is given by Nu =
hL = f (P r)Gr1/4 , λair
where L = 1 m is the plate size in the vertical direction. Here Gr is the Grashof number gβ∆T L3 Gr = , ν2 where g is the gravity, β is the thermal expansion coefficient, ∆T = Ts − Ta , ν is the kinematic viscosity of air. We want to relocate our company on the Moon to take advantage of reduced convective transfers (loss), and so reduce the heating to maintain Tp .
Heat Transport and Transfer
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Heat transport Mechanisms Heat transport equation Heat transfer at interfaces Exercises
Double glazing Insulation of a steam duct Sticking plates on the Moon (2010 exam #1)
Sticking plates on the Moon
Compute the ratio hM oon /h (we suppose ∆TM oon = ∆T ). Compute hM oon . Compute the corresponding heat flux qM oon . Compute the temperature we need to impose on the plate Tp M oon . Explain the small difference between Tp and Tp M oon , knowing that gravity, and thus Gr have been divided by 6 !
Heat Transport and Transfer
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