5 CONVECTION
Incropera Chapters 6-9, Cengel 2011 Chapters 6-7, Welty et al. Chapter 6-9
THIS WEEK:
Fundamental nature of convection, convective heat transfer coefficient (h)
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Forced versus Natural convection
Dimensionless numbers and empirical correlations for
determining convective heat transfer coefficient, h, so that you can solve Newton’s law of cooling for a range of problems
Learning objectives: By the end of week 5 should be able to:
Define and explain the significance of the Nusselt, Reynolds and Prandtl numbers for heat transfer
Explain the difference between forced and natural convection, and draw temperature and velocity boundary layer profiles for both cases
Calculate and apply heat transfer coefficients (h, Wm-2K-1) for forced (internal and external) and natural convection using empirical relationships for common geometries, and use these to calculate rate of convective heat transfer;
Describe and quantify the effects of changes in fluid properties, velocity and geometry on heat transfer coefficients
DEFINITION
Convection is heat transfer from a solid surface to a fluid.
It is the most common mode in chemical/environmental engineering. Rate of Heat transfer: Newton’s Law:
q hAT ST
q = rate of heat transferred between surface and surrounding fluid
(units: W)
h = heat transfer coefficient (W/m2K)
TS = surface temperature (units: K)
T∞ = ambient fluid temperature (units: K)
What determines the magnitude of h? pin e
Pla ste rb oar d
Convective heat transfer is a function of the type of flow.
Forced convection – Flow generated by external means (pump or fan).
Gases: h = 25 – 250 W/m2K
Liquids: h = 50 – 20,000 W/m2K
Natural convection – Flow induced by buoyancy forces due to temperature generated density differences.
Gases: h = 2 – 25 W/m2K
Liquids: h = 50 – 1,000 W/m2K
Boiling & condensation convection – Phase-change induced flow creates latent heat transfer and fluid mixing.
Gases & Liquids: h = 2,500 – 100,000 W/m2K e.g.
What you already know about h:
HOW CONVECTION WORKS: FORCED CONVECTION
BOUNDARY LAYERS
Fig 6.6 Velocity Boundary layer development on a flat plate (Incropera & DeWitt, Bergman & Lavine, 2007)
DIMENSIONLESS GROUPS
h values are most reliably calculated using empirical expressions.
For forced convection, Nu = f(Re, Pr) (parameters depend on configuration and flow regime, see table at the end of this section for specific correlations)
Alternatives (see text): Colburn j factors: j = f/2 = f(Re, Pr) NUSSELT NUMBER, Nu
𝑁𝑢 = h𝐿𝑐 𝑘
Equation H-1
h = film heat transfer coefficient (W/m2K) Lc = characteristic length (m)
k = average thermal conductivity (W/mK)
Nu values may be local (at a point on the surface) or average (integrated over whole surface).
Q. What is the physical meaning of the Nusselt number?
Q. What are the differences between the Nusselt number and the Biot number?
REYNOLDS NUMBER, Re
𝑅𝑒 = 𝜌𝑢𝐿𝑐 = 𝑢𝐿𝑐 = 𝒊𝒏𝒆𝒓𝒕𝒊𝒂 𝒇𝒐𝒓𝒄𝒆 Equation H-2
𝜇 𝝂 𝒗𝒊𝒔𝒄𝒐𝒖𝒔 𝒇𝒐𝒓𝒄𝒆
Lc = characteristic length (m)
u = average velocity (m/s)
μ, ρ – dynamic viscosity (Pa.s) & fluid density (kg/m3), respectively
Re is the property of fluid & flow conditions. It characterizes flow regimes. For flow in circular pipes, these transitions occur at:
– laminar flow – transition flow
– turbulent flow
What is the physical meaning of the Reynolds number?
PRANDTL NUMBER, Pr
𝑃𝑟 = 𝐶𝑝𝜇 = 𝜈 =𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦
Equation H-3
𝑘 𝛼 𝑡h𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑐𝑖𝑣𝑖𝑡𝑦
k = average thermal conductivity (W/mK) ν = kinematic viscosity (m2/s)
α = thermal diffusivity (m2/s)
It is a property of the fluid:
For gases: Pr ≈ 1, rate of momentum & heat transport across fluid approx. equal, and boundary layer thicknesses similar
For liquid metal: Pr << 1 (k is very large, thick thermal boundary layer)
For oils: Pr >> 1 (very high heat capacity, thin thermal boundary layer)
(Values of Pr for various materials are available in Data Pack)
Q. What is the significance of the Pr for heat transfer?
CONVECTION CORRELATIONS
The correlations are written in the following forms:
μ 0.14 Forced convection Nu C1Rem Prn
NATURAL CONVECTION
μw Natural convection Nu C2 Gr.Prp C2 Rap
CONFIGURATION
Flat surfaces – vertical Flat surfaces –
horizontal Cylinders – vertical Cylinders – horizontal Spheres Inside circular tubes
FORCED CONVECTION
a. b. c. d. e.
Lc for Re, Nu
distance x down plate
diameter d for crossflow
diameter d Inside diameter
2000Re10000
Laminar C1 m n 0.65 1⁄2 1⁄3
0.95 1⁄3 1⁄3
1.0 1⁄3 1⁄3c 1.86 1⁄3 1⁄3d
Turbulent C1 m n
0.69 0.47 1⁄3
0.7 1⁄2 1⁄3 0.023 0.8 1⁄3e
Lc for Gr, Nu
height L height L Smaller dimension L height L height L diameter d diameter d diameter d –
0.59 1⁄4 0.10 1⁄3 0.54 1⁄4 0.27 1⁄4 0.59 1⁄4 0.10 1⁄3 0.53 1⁄4 0.13 1⁄3 0.43 1⁄4c
Restriction
104
Use d/L correction factor to account for higher HT in short
The H value
Factors affecting heat transfer are encapsulated in the h value.
EXAMPLE – FORCED CONVECTION:
A 25 cm diameter stainless steel ball ( = 8055 kgm-3, Cp = 480 J/(kg.K)) is removed from the oven at a uniform temperature of 300 C. The ball is then subjected to the flow of air at 1 atm pressure and 25 C with a velocity of 3 m/s. The surface temperature of the ball eventually drops to 200 C. Determine the average convective heat transfer coefficient during this cooling and estimate how long it will take.
NATURAL CONVECTION
Boundary layer (Cengel 2011 p524)
h values are most reliably calculated using empirical expressions.
For natural convection, Nu = f(Gr, Pr) (parameters depend on configuration and flow regime, see table at the end of this section for specific correlations)
1 V V T
If temperature and pressure are low, can assume ideal gas law (for gas buoyancy).
PV nRT V nRT
P dV nR
1dV P nR1
VdT nRT P T
GRASHOF NUMBER, Gr 3
𝑔𝛽(𝑇 − 𝑇 )𝐿 𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒 𝐺𝑟= 𝑠 ∞𝑐=
𝑣2 𝑖𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
g – acceleration due to gravity (9.8 ms-2) β – buoyancy (K-1)
Ts – surface temperature
T∞ – bulk fluid temperature
Lc- characteristic length scale (depends on geometry) ν- kinematic viscosity (m2s-1 )
Q. What is the physical meaning of the Grashof number?
RALEIGH NUMBER, Ra
Q. What is the physical meaning of the Raleigh number?
HOW TO ACCOUNT FOR VARIATION IN FLUID PROPERTIES WITH TEMPERATURE IN BOTH FORCED CONVECTION AND
NATURAL CONVECTION?
FORCED CONVECTION – evaluate fluid properties at bulk temperature T∞ , and apply viscosity correction
0.14
temperature
Tf TwT 2
NATURAL CONVECTION – evaluate fluid properties at the film
where μ is bulk viscosity, μw is viscosity at the wall.
EXAMPLE – NATURAL CONVECTION:
A 6m-long section of an 8-cm diameter horizontal hot-water pipe passes through a large room whose temperature is 20C. If the outer
surface temperature of the pipe is 70C, determine the rate of heat loss from the pipe by natural convection.
1.1 CONVECTION SUMMARY
Nu=C1 Rem Prn (μ / μw)0.14
Nu=C2 (Gr Pr )p = C2 Ra p
INTERNAL FORCED
Circular pipe
Flow regime:
Laminar Turbulent
Square duct
Flow regime:
EXTERNAL FORCED
Flow over sphere
Flow over sphere
Flow over cylinder
Flow over vertical cylinder
Flow over horizontal cylinder
Laminar Turbulent
1.2 CONVECTION CALCULATIONS AND PROBLEM SOLVING
HOW TO DETERMINE HEAT TRANSFER COEFFICIENT h
1. Determine if convection is FORCED or NATURAL
For FORCED CONVECTION:
2. Definegeometryandinternal/externalconvection and hence length scale for Re, Nu
3. DeterminefluidpropertiesatbulktemperatureT∞
4. CalculateReanddetermineflowregime
5. Selectappropriateparametersforequation: Nu=C1 Re m Pr n (μ /μw)0.14
For NATURAL CONVECTION : 2.Definegeometry
and hence length scale for Gr, Nu
3.Determinefluidpropertiesatfilmtemperature: Tf=0.5 (Tw+T∞)
4.CalculateRa=GrPranddetermineflowregime
5.Selectappropriateparametersforequation: Nu=C2 (Gr Pr) p
6. Calculate Nu
7. Calculate h from Nu
8. Check h against expected range
1.3 EXAMPLES OF CONVECTION PROBLEMS
Fan cooling – forced, external convection;
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1.4 SUMMARY OF COMMON ASSUMPTIONS FOR CONVECTION PROBLEMS
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