CONVECTIVE HEAT-TRANSFER COEFFICIENT Introduction

21 CONVECTIVE HEAT-TRANSFER COEFFICIENT Introduction

In most transport processes heat transfer in fluids is accompanied by some form of fluid motion so that the heat transfer does not occur by conduction alone.

Forced convection: fluid motion arises principally from a pressure gradient caused by a pump or blower

Natural convection: fluid motion arises only from density differences associated with the temperature field

Supercritical Fluid Process Lab Supercritical Fluid Process Lab Introduction

Whether the heat-transfer mechanism is natural or forced convection, the fluid motion can be described by the equations of fluid mechanics.

@ low velocities → the flow is laminar throughout the system @ high velocities → laminar near the heating surface + turbulent some distance away

Although fluid velocities in natural convection are usually lower than in forced convection, it is incorrect to think of natural convection as causing only laminar flow; turbulence dose occur when critical Reynolds number for the system is exceeded.

Supercritical Fluid Process Lab Analytical Solution

All problems of convective heat transfer can be expressed in terms of the differential mass, energy, and momentum balances. However, the mathematical difficulties connected with integration of these simultaneous nonlinear partial differential equations are such that analytical solutions exist only for simplified cases.

One of the problems simplest to deal with analytically is that of Heat transfer between a fluid and a flat plate when the fluid is flowing parallel to the plate. (Fig. 21-1)

Supercritical Fluid Process Lab Heat transfer between a fluid and a flat plate when the fluid is flowing parallel to the plate. (Fig. 21-1)

Edge of thermal boundary layer

t0 t0 t0 t 0 y t t0 0 t >t Direction of flow 0 s x

t t t t x0 S x1 S x2 S x3 S

Leading edge of the plate flat plate Fig. 21-1 Development of a thermal boundary layer for flow over a flat plate

Supercritical Fluid Process Lab Temperature profiles in a developing thermal boundary layer on a flat plate

x0 t0 x Edge of thermal 1 x2 boundary layer x3

t0 t0 t 0 y t0

t, fluid temperature t t t t t s x0 S x1 S x2 S x3 S

flat plate y, distance normal to flat plate

Supercritical Fluid Process Lab SupercriticalSupercritical Fluid Fluid Process Process LabLab Heat transfer system Heat is transferred between a fluid and the wall of a pipe

Edge of thermal boundary layer

tS tS tS tS tS tS tS

t0 Direction of flow t0

t0 t0

tS tS tS tS tS tS tS Thermal-entrance length

Temperature profile become flatter.

Supercritical Fluid Process Lab Supercritical Fluid Process Lab Temperature profiles near the entrance of a pipe

t0 x0 x1 x2 x3

ts Figure 21-4

ri r=0 ri

Supercritical Fluid Process Lab Temperature profiles vs Velocity profiles

The temperature profile at a point in a flow system is influenced by the velocity profile.

The velocity profile is influenced by the temperature profile. (The velocity profile of an isothermal system may differ substantially from the velocity of a system in which heat is being transferred.)

Supercritical Fluid Process Lab Differential energy balance

(8-11)

Navier-Stokes equation

Supercritical Fluid Process Lab Simplification

The non-isothermal velocity profile can be used in solving the differential momentum and energy balances for a non- isothermal system.

This simplification may introduce a serious error when the viscosity of the fluid is strongly dependent on temperature.

Frequently the temperature gradient is greatest near the wall, and it is in this region that the velocity gradient is also greatest. The effect of temperature on the viscosity of the fluid at the wall may therefore have a pronounced effect on both the velocity and temperature profiles of the system.

Supercritical Fluid Process Lab m

t0 t0

ts Supercritical Fluid Process Lab Individual heat transfer coefficient, h h is a function of

the properties of the fluid, the geometry and roughness of the surface, and the flow pattern of the fluid.

Supercritical Fluid Process Lab Evaluation of individual heat transfer coefficient, h

Several methods are available for evaluating h

Laminar flow systems → Analytical methods

Turbulent systems → Integral methods → Mixing-length theory → Dimensional analysis

Supercritical Fluid Process Lab Evaluation of individual heat transfer coefficient, h

⎛ dt ⎞ = kdAdq ⎜ ⎟ 0 −= s )( dAtth ⎝ dy ⎠ y=0

k ⎛ dt ⎞ ⎧ [ s 0 −− ttttd s )/()( ]⎫ h = ⎜ ⎟ = k⎨ ⎬ 0 − tt s ⎝ dy ⎠ y=0 ⎩ dy ⎭ y=0

Velocity of flow past the heated surface ↑ Temperature gradient at the wall ↑ Heat transfer coefficient, h ↑

Supercritical Fluid Process Lab Supercritical Fluid Process Lab Individual film coefficient, h

k Fictitious film h = Δxe Δxe

If the resistance to heat flow is through of as existing only in a Fully developed laminar film, h is k/Δxe where Δxe is t the equivalent thickness of a Turbulent flow stationary film just thick enough to Buffer offer the resistance corresponding zone to observed value of h. Since there is often an appreciable resistance in the turbulent core and since the stationary film has not even an approximate physical counterpart in r Laminar laminar flow, boiling and radiation, sublayer we shall refer h as an individual heat transfer coefficient.

Supercritical Fluid Process Lab Definition of h 1 tb = xtdAu Auav ∫A

Supercritical Fluid Process Lab Supercritical Fluid Process Lab Supercritical Fluid Process Lab Example 21-1

The analysis of unsteady-state heating or cooling of solid objects can be often be simplified by assuming that the resistance to heat conduction inside the solid is negligible compared to the convective heat-transfer resistance in the surrounding fluid.

The circumstances necessary to justify this assumption are a high thermal conductivity for the solid and a low convective heat transfer coefficient in the adjacent fluid. A metallic object being heated or cooled in air often constitute such a system.

Supercritical Fluid Process Lab Example 21-1

To illustrate, we shall calculate the time required for a mercury thermometer initially at 70oF, placed in an oven at 400oF, to reach 279oF. The thermometer bulb has a diameter of 0.24 in and will be assumed to be adequately represented as an infinitely long cylinder with negligible resistance to heat transfer either in the glass which is very thin or in the mercury, which has an adequately high thermal conductivity [k=5.6 Btu/(h)(ft)(oF) at 140oF]. The mean convective heat-transfer coefficient between the outside of the thermometer and the air in the oven will be taken as h=2 Btu/(h)(ft2)(oF).

Supercritical Fluid Process Lab Supercritical Fluid Process Lab = 4.24 min

For an infinite cylinder, A/V=(πDL)/(πD2/4)= 4/D Supercritical Fluid Process Lab = eY ⋅− FoBi

For an infinite cylinder, L=V/A=D/4 θ − = eY τ

τ = min24.4

Supercritical Fluid Process Lab θ

Supercritical Fluid Process Lab Temperature response, t τ =time constant

50 20 10 5 1 0.5

Supercritical Fluid Process Lab Temperature response, t, with time

0.5 1 5 10 20 t τ = 50

θ min,

Supercritical Fluid Process Lab BiotBiot NumberNumber (Bi)(Bi)

hL Bi = ks

internal the internal thermal resistance of a solid Bi = boundary the boundary layer thermal resistance

Supercritical Fluid Process Lab FourierFourier NumberNumber ((FoFo))

αθ Fo = L2

heat of rate the rate of heat conduction F = O the thermalrate energy storage ain solid

Supercritical Fluid Process Lab ReynoldsReynolds NumberNumber (Re)(Re)

Lu Re = ∞ ν

forces inertia the inertia forces Re = forces viscousthe forces

Supercritical Fluid Process Lab PrandtlPrandtl NumberNumber (Pr)(Pr)

C μ ν Pr = p = k α

molecular the molecular momentum Pr = ydiffusivit thermalthe ydiffusivit

Supercritical Fluid Process Lab Individual heat transfer coefficient, h Overall heat transfer coefficient, U

= ΔthAq individual

= ΔtUAq overall

Supercritical Fluid Process Lab hi h0

Supercritical Fluid Process Lab Individual temperature drops

1 21 =− qtt Ah ii

Δrb 32 =− qtt Ak ,lmbb

Δrc 43 =− qtt Ak ,lmcc 1 54 =− qtt Ah 00

Supercritical Fluid Process Lab Overall temperature drops

overall −=Δ ttt 51

1 Δrb Δrc 1 51 qtt +=− q + q + q Ah ii Ak ,lmbb Ak ,lmcc Ah 00 ⎛ 1 Δr Δr 1 ⎞ q⎜ b c ⎟ ⎜ += + + ⎟ ⎝ Ah ii Ak ,lmbb lmcc AhAk 00, ⎠

− tt Δt q = 51 q = overall ⎛ 1 Δr Δr 1 ⎞ ΣR ⎜ b c ⎟ ⎜ + + + ⎟ ⎝ Ah ii Ak ,lmbb lmcc AhAk 00, ⎠

Supercritical Fluid Process Lab Overall coefficients of heat transfer based on the outside area U0 ΔtA q = 0 overall ⎛ A ΔrA ΔrA 1 ⎞ ⎜ 00 b 0 c ⎟ ⎜ + + + ⎟ ⎝ Ah ii Ak ,lmbb lmcc hAk 0, ⎠

1 A ΔrA ΔrA 1 += 00 b + 0 c + U 0 Ah ii Ak ,lmbb lmcc hAk 0,

= 00 ΔtAUq overall

Supercritical Fluid Process Lab Overall coefficients of heat transfer based on the inside area U0 ΔtA q = overalli ⎛ 1 ΔrA ΔrA A ⎞ ⎜ bi ci i ⎟ ⎜ + + + ⎟ ⎝ hi Ak ,lmbb Ak lmcc Ah 00, ⎠

11 ΔrA ΔrA A += bi + ci + i hU ii Ak ,lmbb Ak lmcc 0, Ah i

= ΔtAUq overallii

Supercritical Fluid Process Lab Overall coefficients of heat transfer

= 00 ΔtAUq overall

Δ= tAUq overallii

Supercritical Fluid Process Lab Overall coefficients of heat transfer

Δt q = overall ΣR 11 R ==Σ 00 AUAU ii

00 = AUAU ii

Supercritical Fluid Process Lab Approximation of overall coefficients of heat transfer based on the inside area Ui

11 ΔrA ΔrA A += bi + ci + i hU ii Ak ,lmbb Ak lmcc 0, Ah i

1 Δr Δr 1 >> b c ,, Approximation hi kb c hk 0

11 Δr Δr 1 += b + c + hU ii kb c hk 0

Supercritical Fluid Process Lab Overall coefficients of heat transfer of the flat parallel walls

111 Δx Δx 1 +== b + c + i 0 hUU i kb c hk 0

Supercritical Fluid Process Lab Overall coefficients of heat transfer

Table 21-2

U, Btu/(h)(ft2)(oF)

Stabilizer reflux condenser 94 Oil pre-heater 108 Reboiler (condensing steam to re-boiling water) 300-800 Air heater (molten salt to air) 6 Steam-jacketed vessel evaporating milk 500

Supercritical Fluid Process Lab Fouling in heat exchanger

Supercritical Fluid Process Lab Fouling Coefficients

Fouling: deposits on heat transfer surface

- hard scale Heat thermal conductivity Transfer = (boiler or evaporator) Coefficient thickness of scale -coke Sandblasting, pneumatic cleaning tools, chemical cleaning (oil heater in a refinery)

Thermal conductivity itself may be high - porous deposits But effective thermal conductivity may be almost as low as that of the fluid. (mud, soot, vegetable) Steam (air, hot water) blowing

Supercritical Fluid Process Lab Fouling Coefficients

= d ΔtAhq scale

1 U = o A A Δr A Δr A 1 o o ++ b o + c o + fouling Ah ii Ah idi kb A ,lmb kc , hA olmc hdi

kc kb Outside Inside Outside Inside fluid Outside fluid fluid fluid fluid

hi Outside fluid Supercritical Fluid Process Lab Fouling Coefficients

= d ΔtAhq scale

1 U = outside Inside o A A Δr A Δr A 11 o o ++ b o + c o ++ fouling fouling Ah Ah k A k hhA ii idi b ,lmb c , odolmc h hdo di

kc kb Outside Inside Outside Inside fluid Outside fluid fluid fluid fluid

hi Outside fluid Supercritical Fluid Process Lab Fouling coefficients

Table 21-3

2 o hd, Btu/(h)(ft )( F)

Overhead vapors from crude-oil distillation 1000 Dry crude oil (300-100oF): Velocity under 2 ft/sec 250 Velocity 2-4 ft/sec 330 Velocity over 4 ft/sec 500 Air 500 Steam (non-oil-bearing) 200 Water, Great Lakes, over 125oF 500

Tubular Exchanger Manufacturers Association, New York, 1949 Supercritical Fluid Process Lab Example 21-2

A reflux condenser contains ¾ in. 16-gauge copper tubes in which cooling water circulates. Hydrocarbon vapors condense on the exterior surfaces of the tubes. Find the

overall heat-transfer coefficient Uo. The inside convective coefficient can be taken as 4500 W/m2·K, and the outside coefficient as 1500 W/m2·K.

Approximately fouling coefficients from Table 21-3 are

Btu/(h)(ft2)(oF)

2 Overhead vapors from crude-oil distillation 1000 hd0 = 5700 W/m ·K 2 Water, Great Lakes, over 125oF500 hdi = 2840 W/m ·K

Supercritical Fluid Process Lab Example 21-2 Find Uo

¾” 16-gauge copper tube outside h =5700 inside do hdi =2840 ri=0.0157 fouling fouling

ro=0.0191

k Outside Cooling Inside fluid Outside ho=1500 water fluid fluid h =4500 Outside i Condensing fluid vapor

unit: W/m2·K

Supercritical Fluid Process Lab Fouling Coefficients

1 U = o A A Δr A Δr A 11 o o ++ b o + c o ++ Ah ii Ah idi kb A ,lmb kc , hhA odolmc outside Inside fouling fouling h hdo di

kc kb Outside Inside Outside Inside fluid Outside fluid fluid fluid fluid

h Outside i fluid Supercritical Fluid Process Lab Example 21-2 2 2 i = 4500 0 =⋅ /1500,/ ⋅ KmWhKmWh hdo hdi 2 hd =×= /5700678.51000 ⋅ KmW k 0 h =×= /2840678.5500 2 ⋅ KmW hi di ho

= rKmWk i =⋅ rm 0 = 0191.0,0157.0,/380 m − rr − 0157.00191.0 r = 0 i = = 0175.0 m lm r 0191.0 ln 0 ln r 0157.0 1 i U = 0 r r ⋅Δ rr 11 r 0191.0 00 ++ 0 ++ 0 = = 00027.0 rh rh hhrk i dlmbidii 0 0 rh ii )0157.0)(4500( r 0191.0 0 = = 00043.0 rh idi )0157.0)(2840( ⋅Δ rr )0191.0)(00165.0( 0 = = 0000047.0 rk lmb )0175.0)(380( 11 11 == ,00018.0 == 00067.0 h 5700 h 1500 d0 0 Supercritical Fluid Process Lab Example 21-2

hdo hdi 1 k U = hi 0 r r ⋅Δ rr 11 ho 00 ++ 0 ++ rh rh hhrk i dlmbidii 0 0 1 = ++ ++ 00067.000018.00000047.000043.000027.0

Resistance of metal wall is negligible

1 Fouling Resistance is significant~39% == /645 2 ⋅ KmW 00155.0

= AUAU ooii 2 U i = = /785)0157.0/(()0191.0)(645( ⋅ KmW

Supercritical Fluid Process Lab Heat Exchangers

Supercritical Fluid Process Lab Shell–and–Tube Heat Exchangers

Supercritical Fluid Process Lab Heat Exchangers

Supercritical Fluid Process Lab Crossflow heat exchangers

(a) (b)

(a) Finned with both fluid unmixed (b) Unfinned with one fluid mixed and the other unmixed

Supercritical Fluid Process Lab Compact heat exchangers

Supercritical Fluid Process Lab Double-pipe heat exchanger

Cold Fluid in

Wc lb/hr tc Differential section

Hot Hot Fluid Fluid Out In

Wh lb/h Wh lb/hr “1” t “2” tc h Cold Fluid out The choice of path for each fluid depends on Wc lb/hr considerations such as corrosion, plugging, fluid pressure, and permissible pressure drop. Supercritical Fluid Process Lab Double-pipe heat exchanger t

Δt1

Δt

Δt2

Δt= Δt1= Δt2= constant

Supercritical Fluid Process Lab Double-pipe heat exchanger t t

Supercritical Fluid Process Lab Double-pipe heat exchanger

Cold Fluid in W lb/hr c t c Differential section Hot Hot Fluid Out Fluid In

Wh lb/h Wh lb/hr “1” “2” tc th Cold Fluid out

Wc lb/hr The determination of the required heat transfer area is one of the principal objective in the design of heat exchangers. For a differential segment of the

exchanger for which the outside tube area is dA0,

= 00 ΔtAUq overall qAdq 0 Generally, = dA Δ= Δ← tUdAtUdq varingare& 0 0 overall 0 0 ∫∫000 ΔtU overall Δ= tdAU ii (21-23) Supercritical Fluid Process Lab Double-pipe heat exchanger

qAdq 0 = dA Outside heat 0 transfer area, A ∫∫000 ΔtU overall 0

The overall coefficient The difference between the mixing-cup temperatures of the of heat transfer U0 hot and cold fluids, Δt

Under steady conditions, the mixing-cup temperature of the hot and cold fluids in a heat exchanger are assumed to be fixed at any cross section normal to the flow. The overall temperature difference is Δt= th -tc.

Supercritical Fluid Process Lab Countercurrent vs. Concurrent

Cold Fluid in

Wc lb/hr

Hot Hot Countercurrent Fluid Out Fluid In W lb/h Wh lb/hr h “1” “2” Cold Fluid out

Wc lb/hr

Cold Fluid in

Wc lb/hr

Hot Hot Concurrent Fluid Out Fluid In W lb/h W lb/hr h (Parallel-flow) h “1” “2” Cold Fluid out

Wc lb/hr

Supercritical Fluid Process Lab Double-pipe heat exchanger Countercurrent

If Cp=constant

Hot fluid, th Hot fluid, th

Cold fluid, tc Cold fluid, tc

Δt2 =Δ − ttt ch Δ = − ttt ch Δt2 Δt Δtd )( 1 Δt1 Mixing-cup temperature Mixing-cup temperature dq

“1” Length “2” “1” q “2”

Supercritical Fluid Process Lab Heat balance for the differential section

Cold Fluid in W lb/hr c tc Differential section

dA0 Hot Hot Fluid Fluid Out dq In

Wh lb/h Wh lb/hr

“1” t “2” tc h Cold Fluid out

Wc lb/hr

= = hphhcpcc = 0 − ch )( dAttUdtCwdtCwdq 0

Supercritical Fluid Process Lab Double-pipe heat exchanger

If Cp=constant The difference between the mixing cup temperatures, Δt, is also linear w.r.t. q. Hot fluid, th

Cold fluid, tc

=Δ − ttt ch Δt2 Δtd )( Δ − Δtt 12 Δt Δtd )( = Mixing-cup temperature 1 dq dq q

“1” q “2”

Supercritical Fluid Process Lab Heat Balance in the differential section of the exchanger

= = hphhcpcc = 0 − ch )( = ΔtdAUdAttUdtCwdtCwdq 000 (21-25)

Δtd )( Δ−Δ tt = 12 (21-26) dq q

Supercritical Fluid Process Lab Heat Balance in the differential section of the exchanger

Δtd Δtd )()( Δ − Δtt 12 = ΔtdAUdq 00 = = (21-27) dq ΔtdAU oo q

Δt A 2 Δtd )( Δ−Δ tt 12 o = dAo (21-28) ∫Δt ∫0 1 oΔtU q

If Uo=constant

⎡ Δ−Δ tt ⎤ 12 LMTD = AUq oo ⎢ ⎥ (21-29) ()/ln ΔΔ tt Logarithmic ⎣ 12 ⎦ Mean Temperature Difference Supercritical Fluid Process Lab Heat Exchange in double pipe heat exchanger

⎡ Δ−Δ tt 12 ⎤ = AUq oo ⎢ ⎥ Δ=Δ= tCwtCw hphhcpcc ⎣ ()/ln ΔΔ tt 12 ⎦

Supercritical Fluid Process Lab Example 21-3

If the temperature of approach (minimum temperature difference between o fluids) is 20 F, determine the Ao, wh for (a) concurrent flow and (b) countercurrent flow. Kerosene 450oF

countercurrent flow

Crude oil W =2000 lb/hr 200oF c 90oF “1” “2”

2 o Overall coefficient Uo=80 Btu/(h)(ft )( F) Specific heat of crude oil = 0.56 Btu/(ib)(oF) Specific heat of kerosene = 0.60 Btu/(ib)(oF) Supercritical Fluid Process Lab Example 21-3 (a) Concurrent flow

Total heat load:

= tCwq cpcc =Δ − = /000,123)90200)(56.0)(2000( hBtu

q 000,123 wh = = = /891 hlb ΔtC hph − )220450)(60.0( The temperature distribution q 450 Ao = ⎡ Δ−Δ tt 12 ⎤ U o ⎢ ⎥ kero ⎣ ΔΔ tt 12 )/ln( ⎦ sene Δt1 = 360 000,123 220 = 200 oil ⎡ − 20360 ⎤ Crude 80⎢ ⎥ 90 The approach=20oF ⎣ )20/360ln( ⎦ Δt = 20 = 1.13 ft 2 2

Supercritical Fluid Process Lab Example 21-3 (b) Countercurrent flow

Total heat load:

tCwq cpcc =Δ= − = /000,123)90200)(56.0)(2000( hBtu

q 000,123 wh = = = /603 hlb ΔtC hph − )110450)(60.0( The temperature distribution q A = o 450 ⎡ Δ−Δ tt 12 ⎤ U o ⎢ ⎥ ΔΔ tt 12 )/ln( ene Δt = 250 ⎣ ⎦ ros 2 Δt1 = 20 ke 000,123 200 = oil ⎡ − 20250 ⎤ 110 Crude 80⎢ ⎥ 90 ⎣ )20/250ln( ⎦ The approach=20oF = 9.16 ft 2

Supercritical Fluid Process Lab If Uo is a function of temperature o = + ΔtbaU

Δt A Δt 2 Δtd )( Δ − Δtt 12 o 2 Δtd )( Δ − Δtt 12 (21-30) = dAo → = Ao ∫Δt 0 ∫∫ Δt 1 oΔtU q 1 )( ΔΔ+ ttba q

Δt2 Δtd )( Δt2 ⎛ a − //1 ab ⎞ = ⎜ + ⎟ Δtd )( ∫Δt ∫Δt 1 )( ΔΔ+ ttba 1 ⎝ Δt Δ+ tba ⎠

Δt ⎡1 1 ⎤ 2 ln t Δ+−Δ= tba )ln( ⎢a a ⎥ ⎣ ⎦ Δt1 Δt ⎡1 Δt ⎤ 2 = ln ⎢a Δ+ tba ⎥ ⎣ ⎦ Δt1

1 ⎛ Δt2 ⎞⎛ Δ+ tba 1 ⎞ = ln⎜ ⎟⎜ ⎟ a ⎝ Δ+ tba 2 ⎠⎝ Δt1 ⎠

1 o ΔtU 21 = ln (21-31) a o ΔtU 12 Supercritical Fluid Process Lab If Uo is a function of temperature o = + ΔtbaU

o2 = + ΔtbaU 2

o1 Δ+= tbaU 1

oo 12 Δ−Δ=− ttbUU 12 )(

−UU b = oo 12 Δ−Δ tt 12

o o Δ−Δ tUtU 1221 a = (21-32) Δ−Δ tt 12

Supercritical Fluid Process Lab If Uo is a function of temperature o = + ΔtbaU

Δ − Δtt 12 1 o ΔtU 21 Ao = ln q a o ΔtU 12

Δ − ΔtUtU a = o o 1221 Δ−Δ tt 12

o Δ − o ΔtUtU 1221 q = Ao ΔtU (21-34) ln o 21 o ΔtU 12

Supercritical Fluid Process Lab Heat transfer rate in double pipe heat exchanger

Δ − Δtt = AUq 12 o = constU oo ⎛ Δt2 ⎞ (21-29) ln⎜ ⎟ ⎝ Δt1 ⎠

o Δ − o ΔtUtU 1221 o = + ΔtbaU q = A ΔtU o (21-34) ln o 21 o ΔtU 12

Supercritical Fluid Process Lab Homework

PROBLEMS 21-1 21-6 21-10 21-15 21-17

Due on November 2

Supercritical Fluid Process Lab