3. HEAT EXCHANGERS

Geometry: Counter flow, parallel flow, cross flow, shell and tube, mixed, unmixed

Basic equations: )TT(C)TT(Cq i,co,cco,hi,hh

)TT(CTCq i,ci,hminmaxminmax

tothhcc R

1AUAUUA

oooo

o,fw

io

i,f

iotot hA

1

A

"RR

A

"R

hA

1R

ft

fo 1

A

AN1

N = number of fins; Af = area of each fin; At = total area (fins + base); At = Ai or Ao

LMTD method

lmTUAq

)T/Tln(

TTT

12

12lm

(parallel and counter flow exchangers)

CF,lmlm TFT (other exchangers: cross flow, shell and tube)

F from Figures 11S.1 to 11S.4

-NTU method

maxqq

)Cr,NTU(f - Tables 11.3 – 11.4 and Figures 11.10 – 11.15, p. 724-727

minC

UANTU

max

min

C

CCr (typically)

Heat transfer from extended surfaces Temperature distribution, T(x), and qf for uniform fins from Table 3.4, p. 161 Fin efficiency from Figures 3.19 and 3.20 and Table 3.5, p. 166-169

qt

fff hA

1R

finno,bfinno,b hA

1R

bT T

4. CONVECTION HEAT TRANSFER

Laminar and turbulent boundary layers (velocity, thermal and concentration) TThAq s

,As,AmA Ahm

For water:

sg

s,A T@v

1 ,

T@vg

,A ,

= relative humidity, and vg = specific volume from Table A.6, p.1006.

Internal flow: /DURe mD and 2300Re c,D

External flow: /xURex and 5c,x 10x5Re

Dimensionless variable defined in Table 6.2, p.408-409

Heat and Mass Transfer Analogy nm PrReC

k

hLNu ; /Pr

nm

AB

m ScReCD

LhSh ; ABD/Sc A = gas being transferred; B = carrier gas

n1p

m

LeCh

h ; ABD/Le for air-water mixtures Le~1 and p

m C

hh

(Lewis Relation)

properties of air

Transition TurbulentLaminar

xc

U∞

Transition TurbulentLaminar

xc

U∞

TThAq btot

finno,bft qqq TThANAhq bfinno,bfft

ft

fo 1

A

AN1

External convection flows (Chapter 7) Methodology:

1. Geometry i. flat plate (7.30, 7.38) ii. cylinders(s) (7.53, 7.58, 759) iii. sphere (7.57) iv. compact HE

tubes circular finned tubes (Webb and Nae-Hyun, 2005 or Figure 11S.5) tubes with plate fins (Gray and Webb, 1988 or Figure 11S.6)

2. Properties at reference temp. (Tref)

Typical: 2

TTTT s

fref

Some use: Tref=T and /s or Pr/Prs Some use: Tref = (Ti + To)/2

3. Determine Reynolds number 4. Select appropriate correlation (Table 7.7, p.484-485)

What if we don’t know correct Tref? Note this in the solution, use some other T (e.g., T), solve problem, calculate new Tref and compare to original, iterate in practice if important (mention procedure in this class, but don’t iterate) Ts=constant vs qs=constant Correlations for constant surface temperature can be applied with constant heat flux when the flow is turbulent

Internal convection flows (Chapter 8)

Methodology A (determine To for a given L) Methodology B (determine L to get a certain To) 1. Geometry: P/A4D ch , )P/(m4/DuRe hmDh

2. Boundary Conditions: Ts=constant or qs=constant (if laminar flow)

3. Properties at appropriate reference temperature (typical: Tref = (Ti+To)/2 Nu ) 4. Calculate Re and entry lengths (xfd,h and xfd,t) 4. Calculate Re and assume fully developed flow

5. Select appropriate correlation (Summary in Table 8.4 p. 567 and Table 8.1, p. 553) 6. Calculate outlet temperature (To) 6. Calculate required length (L)

(i) Ts=constant:

hCpm

PLexp

TT

TT

T

T

i,ms

o,ms

i

o

(ii) qs=constant:

Cpm

PLqTT s

i,mo,m

7. Total heat transfer: lmTAhq and TCpmq

8. Evaluate assumption of Tref for properties In this class, note difference and how you would solve but don’t iterate. In practice, iterate if important.

8. Evaluate assumption of fully developed flow In this class, note difference and how you would solve but don’t iterate. In practice, iterate if important

Mass transfer relations for step 6: T → , h → mh , Cpm → BB /m

BB

m

i,m,As,A

o,m,As,A

i,A

o,A

/m

hPLexp

i,m,Ao,m,A

B

BA

mm

Fully developed regionThermal entrance region

t

r

xfd,t

T(x,r)=T(0,r)=constant

ro

Ts = constant qs” = constant

Fully developed regionThermal entrance region

t

r

xfd,t

T(x,r)=T(0,r)=constant

ro

Ts = constant qs” = constant ReD,c2300

Laminar: ; ;

Turbulent:

lm,AmA Ahm i,Ao,A

i,Ao,Alm,A ln

0312.0s

1

l

t328.04N d

t

S

SRe14.0j

0.110.20.319-

d1/3d t

s

e

sRe 0.134=

PrRe

Nu=j

Equation valid for the following conditions: 1,100 Red 18,000 0.13 s/e 0.63 1.0 s/t 6.6 0.09 e/d 0.69 0.01 t/d 0.15 1.5 ST/d 8.2

Natural (free) convection and combined natural and forced convection Grashof number

2

3s

L

L)TT(gGr

where ]K/1[T/1 for gases (from property tables for liquids)

Rayleigh number

3

sLL

L)TT(gPrGrRa

9c 10Re

natural and forced convection

1

Re

Gr2

L

L (0.2 – 5)

nN

nF

n NuNuNu

general: n=3 traverse flow: plate (n=7/2) or

sphere, cylinder (n=4)

natural

1

Re

Gr2

L

L

Nu=f(Ra, Pr)

forced

1

Re

Gr2

L

L

Nu=f(Re, Pr)

Methodology 1. Geometry (plate, cylinder or sphere) 2. Properties at reference temp: Tref = (Ts +T)/2

3. Determine 2LL Re/Gr

4. Select appropriate correlation(s) - Table 9.2 (p. 617-618)

5. RADIATION HEAT TRANSFER

Engineering properties for materials at different T [Table A.11] , for solar radiation [Table A.12]

Radiation exchange between bodies View Factor F12 – fraction of the radiation leaving surface 1, which is intercepted by surface 2

reciprocity relation 221112 AFAF

summation rule 1FN

1jij

Two surface enclosures (diffuse, gray and opaque surfaces) Diffuse: independent of angle Gray surface: = Kirchoff’s Law: = for all surfaces in an enclosure

Combined radiation, conduction and convection heat transfer

Fij is given for common geometries in Tables 13.1 - 13.2 and Figures 13.4 – 13.6, p. 865-869

qconv,1 = h1A1(T-T1) qconv,2 = h2A2(T-T2)

q12

Rrad

Some special enclosures given in Table 13.3, p. 885

Rrad U, T

A1,T1, 1 T3

22

2

11211

1rad A

1

AF

1

A

1R

rad

42

41

rad

2b1b12 R

TT

R

EEq

rad

2b1brad R

EEq

cond

13cond R

TTq