Modes of Heat Transfer

Review of Heat Transfer & Fluid Flow

Dr. Md. Zahurul Haq

Professor
Department of Mechanical Engineering
Bangladesh University of Engineering & Technology (BUET)
Dhaka-1000, Bangladesh

zahurul@me.buet.ac.bd
http://zahurul.buet.ac.bd/

ME 307: Heat Transfer Equipment Design T946

http://zahurul.buet.ac.bd/ME307/

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Heat Conduction Through Plain Wall

k(T ) = k0 (1 + βk T )

T944

Temperature distribution for steady-state T945

conduction through a plane wall, and analogy
between thermal and electrical circuits. Variable thermal conductivity T949

Q̇cond = kA ∆T ∆T ∆T ∆x 1 Variation of thermal conductivity. (a) gases and (b) liquids.

∆x = UA∆T = 1/UA = Rcond :→ Rcond = kA = UA
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 Heat Convection Non-dimensional Numbers in Heat Convection

hL
≡ Nu = f (Re, Pr , · · · )
k

• Reynolds number, Re ≡ ρVL inertia force

µ = viscous force
ν cp µ molecular diffusivity of momentum
• Prandtl number Pr ≡ α = k = molecular diffusivity of heat
• Nusselt number Nu ≡ hL
T947 k
• Peclet number Pe ≡ RePr
(a) Forced convection. (b) Natural convection.
(c) Boiling. (d) Condensation. • Graetz number, Gz ≡ Pe dx
T948
(Ts −T∞ ) (Ts −T∞ ) 1
Q̇conv = hA(Ts − T∞ ) = 1/hA = Rconv :→ Rconv = hA

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Conduction + Convection

Heat Radiation Heat Conduction + Convection through Plain Wall

Q̇rad = ǫσA(T24 − T14 )

Q̇rad = hr A(Ts − Tsur ) : hr ≡ ǫσ(Ts + Tsur )(Ts2 + Tsur

2
)

T954

Radiation exchange: (a) at a surface and (b) between a surface and large T950

surroundings. (a) Temperature distribution. (b) Equivalent thermal circuit.

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 Conduction + Convection Conduction + Convection

Heat Conduction + Convection: Hollow Cylinder

T952
T951

Hollow cylinder with convective surface conditions.

Equivalent thermal circuit for a series composite wall.
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Conduction + Convection Heat Transfer Correlations

Simplified Equations for Air: Free Convection

T661
T953

Temperature distribution for a composite cylindrical wall.

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 Heat Transfer Correlations Heat Transfer Correlations

Flow inside Circular Duct

Sieder & Tate: [Laminar flow]

 0.14
µb RePr Holman Ex. 10-1 ⊲ Hot water at 98o C flows through a 2-in schedule 40
Num = 1.86(Gz)1/3 ; Gz =
µw L/D horizontal steel pipe [k = 54 W/mo C] and is exposed to atmospheric air at 20o C.
The water velocity is 25 cm/s. Calculate the overall heat transfer coefficient for
0.14
[Uo = 7.84 W/m2o C]

0.48 < Pr < 16700; 0.044 < µb /µw < 9.75; (Gz)1/3 µb
>2 this situation, based on the outer area of pipe.
µw

Dittus-Boelter: [Turbulent flow]

0.4 : heating
Num = 0.023Re 0.8 Pr n ; n=
0.3 : cooling

0.7 < Pr < 16700; Re > 10000; L/D > 60

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HTX fouling HTX fouling

Heat Exchanger: Fouling

During operation, heat exchangers become fouled with an accumulation of

deposits of one kind or another on the heat transfer surface areas. Major
categories of fouling:
1 scaling or precipitation fouling
2 particulate fouling
3 chemical reaction fouling
4 corrosion fouling
5 biological fouling
6 solidification fouling T1013

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 HTX fouling HTX fouling

Cengel Ex. 11-2 ⊲ Effect of fouling on overall heat transfer coefficient.

Double-pipe (shell-and-tube) heat exchanger made of steel, k = 15.1 W/m-K,
inner-diameter of outside shell is 3.2 cm.
[Uo = 315 W/m2o C, Uo,new = 389.5 W/m2o C]

T1248

Q̇ = Uo Ao ∆TLM
1 1 1 Rd,i ln(Do /Di ) Rd,o 1
Uo Ao = Ui Ai =R = hi Ai + Ai + 2πkL + Ao + ho Ao
1 1 1 1 1
Uo ≃ hi + ho ; U ≃ Uo + Rd,i + Rd,o

T704

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Condensation & Condensers Condensation & Condensers

Condensation Correlations Condensation on inclined surfaces: [Laminar (Re <1800)]

Ahm (Tv −Tw ) 4ṁ
Rate of condensation, ṁ = hfg ; Re = µl P

g ρl (ρl − ρv )hfg kl3
1/4
hm = 0.943 sin ϕ
µl (Tv − Tw )L
hfg evaluated at Tv

 Condensation on horizontal tube: [Laminar (Re <1800)]

 πD for vertical tube of outside diameter D
Wetted perimeter , P = 2L for horizontal tube of length L 1/4
g ρl (ρl − ρv )hfg kl3

 hm = 0.725
w for vertical or inclined plate of width w µl (Tv − Tw )D
Condensation on vertical surfaces: [Laminar (Re <1800)]
Comparison of vertical tube of length L and horizontal tube of diameter D:
1/4
ρv )hfg kl3

g ρl (ρl −  1/4
hm = 1.2×0.943 hm,vert D
µl (Tv − Tw )L = 1.56
hm,horz L
• Properties are evaluated at film temperature, Tf = 21 (Tw + Tv ) If L = 100D → hm,horz ≃ 2.0hm,vert . For condensation, horizontal tube
• Flow is laminar if Re < 1800. arrangements are generally preferred.
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 Condensation & Condensers Condensation & Condensers

Condensation on horizontal tube banks:

1
hm |N tubes = hm |1 tube
N 1/4
Ozikik Ex. 10.1 ⊲ Air free saturated steam at 65o C, P = 25.03 kPa condenses on
the outer surface of a 2.5 cm OD, 3-m long vertical tube maintained at constant
Condensation inside horizontal tube:
temperature of 35o C by the flow of the cooling water. Estimate average heat
"
′ k 3 1/4
# transfer coefficient and the rate of condensate flow at the bottom of the tube.
g ρl (ρl − ρv )hfg l
hm = 0.555 [ṁV = 0.01124 kg/s]
µl (Tv − Tw )D
Ozikik Ex. 10.2 ⊲ Redo Ozisik Ex. 10.1 assuming horizontal tube.
ρ v uv D
′
hfg = hfg + 83 cp,l (Tv − Tw ); Rev = µv 6 35000 [ṁH = 0.02386 kg/s]

Turbulent film-wise condensation: vertical tube

1/3
µ2l

hm = 0.0077Re 0.4
kl3 ρ2l g

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Condensation & Condensers Flow inside Tubes/Ducts

Flow Parameters: Flow inside Tubes/Ducts

Dewitt Ex. 10-4 ⊲ The tube bank of a steam condenser consists of a square
array of 400 tubes, each 6 mm in diameter. If horizontal, unfinned tubes are • Turbulent flow inside circular duct, if Ref > 2300.
exposed to saturated steam at a pressure of 0.15 bar and the tube surface is • Pressure drop, ∆P = f DL ( 12 ρum
2)
maintained at 25o C, what is the rate at which steam is condensed per unit length • Pumping power, Ẇ = V̇ ∆P = Vav A∆P
of the tube bank?
[ṁ = 0.474 kg/s] ξcorr [64/Ref ] for laminar flow
• f =
(1.82 log10 Ref − 1.64)−2 for turbulent flow

Ref ≡ ρumµDh = Reynolds number for friction analysis

ReQ ≡ ρumµDe = Reynolds number for heat transfer analysis
f = (Darcy) friction factor
Dh ≡ 4A/Pwetted = hydraulic diameter
De ≡ 4A/Pheat = equivalent diameter
T721
um = mean flow velocity
h i
De
⇒ ReQ = Ref D h

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 Flow inside Tubes/Ducts Flow inside Tubes/Ducts

012
312
Penoncello Ex. 5-10 ⊲ The outer pipe of a double pipe heat exchanger is 4-in
schedule 40 commercial steel. The inside tube is 3 std type M copper. Water at
an average temperature of 30o C is flowing in the annulus at a rate of 10 m3 /h.

314
Determine the Reynolds numbers of the annular flow for use in hydraulic analysis
and heat transfer analysis.
T761 [Ref = 2.43 × 105 , Ree = 5.56 × 105 ]
• Pipe area:
• De = Dh = IDp
• ∆P = f IDL ( 21 ρp Vp2 ), ξcorr = 1
p
• Annular area:
2 2
−ODp
• Dh = IDa − ODp , De = IDaOD
p
2
• ξ 1 = (1−κ)
1+κ 1+κ
2 + (1−κ) ln(κ) , κ = ODp /IDa
corr
• ∆P = (f DL + 1)( 12 ρa Va2 )
h

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