SUMMARY TK2203 – HEAT TRANSFER OPERATION

2nd CHAPTER : HEAT TRANSFER BY CONVECTION Convection coefficient heat transfer resistance in fluids, also referred to as film coefficient because it simply refers to thermal resistance of a stagnant thin-film on a surface are in which heat transfer ensues.

Physical symbol for film coefficient ℎ[=]

There are two types of convection: 1. Forced convection fluids’ moving driven by external force, i.e. pump, fan, or such. 2. Natural convection fluids’ moving driven by density difference due to temperature gradient

Special hallmark : Forced convection case fluid is moving with certain velocity

WHILE natural convection case there’s no information about fluid’s velocity General equation for convection : Heat flux occurred by convection (heat rate Q / surface area A), Q/A, is proportional to temperature difference between fluids and complementary surface:

퐍퐞퐰퐭퐨퐧 퐬퐥퐚퐰퐨퐟퐜퐨퐨퐥퐢퐧퐠 ∶ 푄퐴 = 푞 = ℎ∆푇

where 푄 = heatrate = W ; A = heattransfersurfacearea(m ), 푞 = heatflux ;

∆푇 = Temperaturedifferencebetweenfluidandsurface(K) h depends on surface geometry and flow condition (regime). MAIN OBJECTIVE IN THIS CHAPTER IS TO FIGURE OUT h ! In order to determine h, it can be conducted by experiments which finally resulted empirical models. Dimensionless number related to h :

1. Reynold number determinant of flow regime (laminar or turbulent)

푹풆 =흆풗ퟐ

흁풗/푳 =흆풗푳흁 ; forpipe, Lcanbesubstitutedto퐷

2. Nusselt number

푵풖 =풉푳풌 =

풒푳풌∆푻 ; forpipe, Lcanbesubstitutedto퐷

3. Prandtl number refering to relative width of hydrodynamic layer to thermal boundary layer

푷풓 =풗ҝ =

흁흆

(풌/흆푪풑) =흁푪풑풌

4. Grashoff number

푮풓 =휷품흆ퟐ푳ퟑ∆푻

흁ퟐ

Note : Red-coloured equation refers to forced convection while green one refers to natural convection. Purple one holds for both.

FORCED CONVECTION For laminar flow in horizontal pipe ( 푁 < 2100 )

(푁 ) =ℎ 퐷푘 = 1,86

푁 푁 퐷퐿

휇휇

,

All physical properties are evaluated at BULK fluid temperature EXCEPT 휇 at wall temperature

For turbulent flow in horizontal pipe ( 푁 > 6000; 0,7 < 푁 < 16000; < 60 )

(푁 ) =ℎ 퐷푘 = 0,027푁 , 푁

휇휇

,

All physical properties are evaluated on BULK fluid temperature EXCEPT 휇 at wall temperature If bulk fluid temperature is not equal to inlet and outlet temperature then evaluate it on arithmatical mean temperature. Sometimes we need to use trial and eror method : h TW µW

at wall temperature For transitional flow in horizontal pipe (2100 < 푁 < 6000 ) , use diagram below:

h value for non-circular conduits can be evaluated by using equivalent diameter (Recall what we learnt in Fluid Mechanics course) Annulus : DEFF = Outer diameter – Inner diameter

effect of L/D

ℎℎ = 1 +

퐷퐿

.

; 2 <퐿퐷 < 20

ℎℎ = 1 + 6

퐷퐿 ; 20 <

퐿퐷 < 60

h is the average value for a tube of finite length L and hL is the value for a very long tube. For liquid metal (i.e. Hg). Peclet number: NPe = NRe . NPr (1) If the flow is turbulent and L/D > 60, 100 < NPe < 10000

푁 =ℎ 퐷푘 = 0.625푁 .

(2) If the wall temperature is constant, L/D > 60 , NPe > 100

푁 =ℎ 퐷푘 = 5 + 0.025푁 .

All physical properties are evaluated at bulk fluid temperature.

Outside various geometries : outside here means the heat transfer occurs on the outside of the body

푁 = 퐶푁 푁

Properties of fluid are evaluated at the film temperature ( Tf ) : 푇 =

Tw : wall temperature and Tb : bulk fluid temperature (1) Parallel flow through flat plate with length L meter

a. Laminar flow : 푁 , < 3 × 10 and 푁 > 0,7

푁 =ℎ퐿푘 = 0,664푁 ,

, 푁 ; 푁 =퐿푣̅휌휇

b. Turbulent flow : 푁 , > 3 × 10 and 푁 > 0,7

푁 =ℎ퐿푘 = 0,0366푁 ,

, 푁 ; 푁 =퐿푣̅휌휇

(2) Perpendicular flow to cylinder axis

푁 = 퐶푁 푁 ; 푁 =퐷푣̅휌휇

with D is outside tube diameter ( can be either outter diameter if the tube is monolayer or outtest diameter if the tube is multilayer). All physical properties are evaluated at the film temperature (Tf). The velocity is undisturbed free stream velocity approaching the cylinder.

(3) Flow past single sphere

푁 = 2 + 0,6푁 . 푁 ; 푁 =퐷푣̅휌휇

; 1 < 푁 < 70000and0.6 < 푁 < 400

Fluid properties are evaluated at the film temperature ( Tf ) (4) Flow past banks of tubes and cylinders

NATURAL CONVECTION For various geometry

(1) Vertical planes and cylinders For L < 1 m ( L means height of vertical surface or plate, or vertical cylinder length) , the following equation holds

푁 =ℎ퐿푘 = 푎

퐿 휌 푔훽∆푇휇

푐 휇푘 = 푎(푁 푁 ) ;푎and푚areconstantgivenbelow.

These equations in the table below are simplified form of equations to calculate h value at 1 atm absolute pressure BUT IT PREVAILS ONLY WHEN (ΔT.L3) < 4.7 m3K AND 255 K < Tf <533 K

The corresponding information to the simplified equations is given in Table 4.7-2 on previous page.

푇 =푇 + 푇

2

NGr : Grashof number ρ : density (kg/m3) µ : viscosity (kg m-1 s-1) ΔT : |TWALL – TBULK| (K) k : thermal conductivity (W m-1 K-1) cP : heat capacity (J kg-1 K-1) β : volumetric coefficient of expansion of fluid (K-1) FYI, for gas: β=1/Tf All physical properties are evaluated at film temperature (Tf) in K

g = 9,80665 ms-2

WARNING: Notice if the absolute pressure is not 1 atm but p pascal, the h value calculated from the table above needs to be multiplied by the following factors:

- If ( 104 < NGr.NPr < 109 ) : ℎ = ℎ ..

- If ( NGr.NPr > 109 ) : ℎ = ℎ ..

(2) In enclosed spaces Two vertical plates of height L m containing the fluid with a gap of δ m

푁 , = ( )and푁 , =

Heat flux : = ℎ(푇 − 푇 )

Enclosed fluid physical properties such as ρ, µ, and k are evaluated on Tf .

푇 =푇 + 푇

2

- If the fluid is gas, the plate is vertical as shown in Figure 1, and L/δ > 3 Calculate the NGr.NPr and then choose one of these satified conditions:

If 푁 , 푁 < 2 × 10 : 푁 , = = 1

If 6 × 10 < 푁 , 푁 < 2 × 10 : 푁 , = 0.2 , = 1

If 2 × 10 < 푁 , 푁 < 2 × 10 : 푁 , = 0.073 ,

- If the fluid is liquid, the plate is vertical as shown in Figure 1, there’s no specific restriction for value of L/δ. Calculate the NGr.NPr and then choose one of these satifying conditions:

If 푁 , 푁 < 10 : 푁 , = = 1

If 10 < 푁 , 푁 < 10 : 푁 , = 0.28 ,

How about liquids or gases in vertical annulus? The same equation holds for case of vertical plates. How about the plates are horizontal plates? If the lower plate is hotter than the upper:

- If the fluid is gas, calculate the NGr.NPr and then choose one of these satified conditions:

If 7 × 10 < 푁 , 푁 < 3 × 10 : 푁 , = 0.21 푁 , 푁

If 푁 , 푁 > 3 × 10 : 푁 , = 0.061 푁 , 푁 - If the fluid is liquid, this equation below holds if 1.5 × 10 < 푁 , 푁 < 10

푁 , = 0.069 푁 , 푁 푁 .

(3) Horizontal cylinders Just have a look at equation for vertical cylinder (number 1) before. Change L in the equation with D (outside diameter of the horizontal cylinder. BE CAREFUL, D IS DIAMETER, NOT RADIUS! Behold table 4.7-1 for horizontal cylinders too!

(4) Horizontal plates The same as (3), but L depends on what sort of plates it is.

- Square plate with side length X : L = X - Two dimensional rectangle with side length X and Y : L = 0.5( X + Y ) - Circular disk with diameter D : L = 0.9D

Figure 1. Enclosed vertical spaces Adapted from Figure 4.7-2 Geankoplis

For the nucleate boiling: (1) Horizontal plate, the equation used to calculate h can be found below:

푞퐴

(btuh ft ) < 5000 ∶ ℎ(btuh ft ℉ ) = 151(∆T(℉))

푞퐴

(kWm ) < 16 ∶ ℎ(Wm 퐾 ) = 1043(∆T(K))

5000 <푞퐴

(btuh ft ) < 7,5 × 10 ∶ ℎ(btuh ft ℉ ) = 0.168(∆T(℉))

16 <푞퐴

(kWm ) < 240 ∶ ℎ(Wm 퐾 ) = 5.56(∆T(K))

(2) Vertical plate 푞퐴

(btuh ft ) < 1000 ∶ ℎ(btuh ft ℉ ) = 87(∆T(℉))

푞퐴

(kWm ) < 3 ∶ ℎ(Wm 퐾 ) = 537(∆T(K))

1000 <푞퐴

(btuh ft ) < 2 × 10 ∶ ℎ(btuh ft ℉ ) = 0.24(∆T(℉))

3 <푞퐴

(kWm ) < 63 ∶ ℎ(Wm 퐾 ) = 7.95(∆T(K))

∆푇 = 푇 − 푇 inKor℉ If the pressure is p atm abs, the vaolue of h at 1 atm calculated above are multiplied py (p/1)0.4. The red box equations are in the natural convection region. What about forced convection boiling inside tubes?

ℎ = 2.55 ∆푇(퐾) 푒 (Wm K )

ℎ = 0.077(∆푇(℉)) 푒 (btuh 푓푡 ℉ ) where p is in kPa or psia (English units).

Condensation of thin film 1. LAMINAR FLOW

For a vertical surface in laminar flow (Re < 1800)

푁 =ℎ퐿푘

= 1,13휌 (휌 − 휌 )푔ℎ 퐿

휇 푘 ∆푇… (1)

ΔT=TSAT - TWALL

L : vertical height of surface or tube hfg : latent heat of condensation in J/kg at TSAT

All phyisical properties except hfg are evaluated at Tf = 0,5(TSAT + TWALL) 2. TURBULENT FLOW Turbulent flow occurs when Re > 1800

푁 =ℎ퐿푘

= 0,0077푔휌 퐿휇

(푁 ) , … . (2)

The solution of that equation is obtained bt Trial and Error method. Use this following equation:

푞 = ℎ퐴∆푇 = ℎ 푚… (3) GENERAL FOR BOTH: Reynold number can be calculated by these following equations:

푁 =4푚휋퐷휇

=4훤휇 ; verticaltube, diamaterD

푁 =4푚푊휇

=4훤휇 ; verticalplate, widthW

m is mass flow rate with unit of kg/s hfg is latent heat of condensation, mathematically defined as: ℎ = ℎ − ℎ = 퐉/퐤퐠 WARNING: Beware of unit of hfg and notice it carefully! It’s casually given in kJ/kg, do not forget to convert it to J/kg by multiplication with 1000. How to do the trial and error that was just mentioned before? 1. Evaluate TSAT, TWALL, and TFILM 2. Evaluate latent heat at TSAT and then evaluate ρLIQUID, ρVAPOR, kLIQUID, µLIQUID at TFILM 3. Assuming the regime whether it’s laminar or turbulent. 4. Calculate h by using either equation (1) (if the initial assumption is laminar flow) or (2) (for the contrary) 5. Calculate area of convection. For instance: in the case of cylinder with outer diameter D with length L and steam

condensing outside its shell, the A is therefore: A=πLD 6. Calculate q by using equation (3), ΔT=TSAT - TWALL 7. Calculate m. 8. Calculate Reynold number. 9. See if the regime is correct as assumed. (critical number=1800) 10. If the regime is not correct as assumed, repeat step 4 for the another one regime.

Figure 2. Film condensation on a vertical plate Adapted from Figure 4.8-3 Geankoplis

1. Knowing the case whether the fluid is one phase or two phases 2. If it appears to be one phase: check whether it’s natural or forced

convection 3. If it appears to be two phases: check whether it’s boiling or

condensation 4. Knowing the geometry of body where heat transfer occurs 5. Notice the restriction of conditions (when it prevails) of all equations

(Reynold num., L/D) 6. Notice at what temperature will the properties of fluid be evaluated

then. 7. Notice if the pressure may affect. 8. BE METICULOUS on putting UNIT of any variables involved in all

related equations.