Objectives

• Heat transfer• Convection • Radiation

• Fluid dynamics• Review Bernoulli equation

• flow in pipes, ducts, pitot tube

Tout

Tin

R1

/A R2

/ARo

/A

Tout

Ri/A

Tin

l1k

1, A

1 k2

, A2

l2

l3

k3,

A3

A2

= A1

(l1

/k1

)/A1

R1

/A1

Tout

Tin

(l2

/k2

)/A2

R2

/A2

(l3

/k3

)/A3

R3

/A3

1. Add resistances for series

2. Add U-Values for parallel

l thickness

k thermal conductivity

R thermal resistance

A area

Convection and Radiation

• Similarity• Both are surface phenomena• Therefore, can often be combined

• Difference• Convection requires a fluid, radiation does not• Radiation tends to be very important for large

temperature differences• Convection tends to be important for fluid flow

Forced Convection

• Transfer of energy by means of large scale fluid motion

In the following text:

V = velocity (m/s, ft/min) Q = heat transfer rate (W, Btu/hr)

ν = kinematic viscosity = µ/ρ (m2/s, ft2/min) A = area (m2, ft2)

D = tube diameter (m, ft) T = temperature (°C, °F)

µ = dynamic viscosity ( kg/m/s, lbm/ft/min) α = thermal diffusivity (m2/s, ft2/min)

cp = specific heat (J/kg/°C, Btu/lbm/°F)

k = thermal conductivity (W/m/K, Btu/hr/ft/K)

h = convection or radiation heat transfer coefficient (W/m2/K, Btu/hr/ft2/F)

ThAQ

Dimensionless Parameters

• Reynolds number, Re = VD/ν

• Prandtl number, Pr = µcp/k = ν/α

• Nusselt number, Nu = hD/k

• Rayleigh number, Ra = …

What is the difference between thermal conductivity and thermal diffusivity?

• Thermal conductivity, k, is the constant of proportionality between temperature difference and conduction heat transfer per unit area

• Thermal diffusivity, α, is the ratio of how much heat is conducted in a material to how much heat is stored

• α = k/(ρcp)

• Pr = µcp/k = ν/α

k = thermal conductivity (W/m/K, Btu/hr/ft/K)

ν = kinematic viscosity = µ/ρ (m2/s, ft2/min)

α = thermal diffusivity (m2/s, ft2/min)

µ = dynamic viscosity ( kg/m/s, lbm/ft/min)

cp = specific heat (J/kg/°C, Btu/lbm/°F)

k = thermal conductivity (W/m/K, Btu/hr/ft/K)

α = thermal diffusivity (m2/s)

Analogy between mass, heat, and momentum transfer

• Schmidt number, Sc

• Prandtl number, Pr

Pr = ν/α

Forced Convection

• External turbulent flow over a flat plate• Nu = hmL/k = 0.036 (Pr )0.43 (ReL

0.8 – 9200 ) (µ∞ /µw )0.25

• External turbulent flow (40 < ReD <105) around a single cylinder• Nu = hmD/k = (0.4 ReD

0.5 + 0.06 ReD(2/3) ) (Pr )0.4 (µ∞ /µw )0.25

• Use with careRe

L = Reynolds number based on length Q = heat transfer rate (W, Btu/hr)

ReD

= Reynolds number based on tube diameter A = area (m2, ft2)

L = tube length (m, ft) t = temperature (°C, °F)

k = thermal conductivity (W/m/K, Btu/hr/ft/K) Pr = Prandtl number

µ∞

= dynamic viscosity in free stream( kg/m/s, lbm/ft/min)

µ∞

= dynamic viscosity at wall temperature ( kg/m/s, lbm/ft/min)

hm

= mean convection heat transfer coefficient (W/m2/K, Btu/hr/ft2/F)

Natural Convection

• Common regime when buoyancy is dominant• Dimensionless parameter• Rayleigh number

• Ratio of diffusive to advective time scales• Book has empirical relations for

• Vertical flat plates (eqns. 2.55, 2.56)• Horizontal cylinder (eqns. 2.57, 2.58)• Spheres (eqns. 2.59)• Cavities (eqns. 2.60)

PrTgHTHgRa

/T 2

33

For an ideal gas

H = plate height (m, ft)

T = temperature (°C, °F)

Q = heat transfer rate (W, Btu/hr)

g = acceleration due to gravity (m/s2, ft/min2)

T = absolute temperature (K, °R)

Pr = Prandtl number

ν = kinematic viscosity = µ/ρ (m2/s, ft2/min)

α = thermal diffusivity (m2/s)

Phase Change –Boiling

• What temperature does water boil under ideal conditions?

Forced Convection Boiling• Example: refrigerant in a tube• Heat transfer is function of:

• Surface roughness• Tube diameter• Fluid velocity• Quality• Fluid properties• Heat-flux rate

• hm for halocarbon refrigerants is 100-800 Btu/hr/°F/ft2

(500-4500 W/m2/°C)

Nu = hm

Di/k

ℓ=0.0082(Re

ℓ2K)0.4

Reℓ = GD

i/µ

ℓ

G = mass velocity = Vρ (kg/s/m2, lbm/min/ft2)

k = thermal conductivity (W/m/K, Btu/hr/ft/K)

Di

= inner diameter of tube( m, ft)

K = CΔxhfg

/L

C = 0.255 kg∙m/kJ, 778 ft∙lbm/Btu

Condensation

• Film condensation• On refrigerant tube surfaces• Water vapor on cooling coils

• Correlations• Eqn. 2.62 on the outside of horizontal tubes• Eqn. 2.63 on the inside of horizontal tubes

Radiation

• Transfer of energy by electromagnetic radiation• Does not require matter (only requires that the

bodies can “see” each other)• 100 – 10,000 nm (mostly IR)

Blackbody

• Idealized surface that• Absorbs all incident radiation• Emits maximum possible energy• Radiation emitted is independent of direction

Radiation emission The total energy emitted by a body,

regardless of the wavelengths, is given by:

Temperature always in K ! - absolute temperatures

– emissivity of surface ε= 1 for blackbody

– Stefan-Boltzmann constant

A - area

4ATQemited

Radiation Equations

2

2

2

1

211

1

42

411

21 111)(

AA

F

TTAQ

2

2

2

1

211

1

3

2

2

2

1

211

1

21

42

41

111

4

111)()(

AA

F

T

AA

F

TTTT

h avgr

tAhQ rrad

Q1-2

= Qrad

= heat transferred by radiation (W, BTU/hr) F1-2

= shape factor

hr = radiation heat transfer coefficient (W/m2/K, Btu/hr/ft2/F) A = area (ft2, m2)

T,t = absolute temperature (°R , K) , temperature (°F, °C)

ε = emissivity (surface property)

σ = Stephan-Boltzman constant = 5.67 × 10-8 W/m2/K4 = 0.1713 × 10-8 BTU/hr/ft2/°R4

Short-wave & long-wave radiation

• Short-wave – solar radiation• <3m• Glass is transparent • Does not depend on surface temperature

• Long-wave – surface or temperature radiation• >3m• Glass is not transparent • Depends on surface temperature

Figure 2.10

• α + ρ + τ = 1 α = ε for gray surfaces

Radiation

Combining Convection and Radiation• Both happen

simultaneously on a surface• Slightly different

temperatures

• Often can use h = hc + hr

Combining all modes of heat transfer

Example of Conduction Convection and Radiation use: Heat Exchangers

Ref: Incropera & Dewitt (2002)

Shell-and-Tube Heat Exchanger

Ref: Incropera & Dewitt (2002)

Fluid Flow in HVAC components

Fundamentals: Bernoulli’s equation Flow in pipes:• Analogy to steady-flow energy equation• Consider incompressible, isothermal flow• What is friction loss?

gDLVf

2

2

2

2VDLfp friction

[ft]

[Pa]

Pitot Tubes

Summary

• Use relationships in text to solve conduction, convection, radiation, phase change, and mixed-mode heat transfer problems

• Calculate components of pressure for flow in pipes and ducts

Any questions about review material?

• Where are we going?• Psychrometrics

• Psychrometric terms• Using tables for moist air• Using psychrometric charts• 7.1 – 7.5, 7.7