Iteration-free calculation of heat transfer coefficients in heat exchangers

The ChemicalE’ngineering Journal, 13 (1977) 233-237 @Elsevier Sequoia %A., Lausanne. Printed in the Netherlands

Iteration-free Calculation of Heat Transfer Coefficients in Heat Exchangers WILFRIED

ROETZEL

Institut fiir Mechanik und Thermodynamik,

Hochschuleder Bundeswehr,~oisten~ofweg85, Hamburg70 (F.R.G.J

(Received 17 May 1976; in final form 25 January 1977)

Abstract If a heat transfer coefficient is dependent on the wall temperature it is usually calculated by an iterative technique. Starting from an earlier iteration-free method which was developed for air-cooled heat exchangers and the cooling of viscous oils, an improved procedure is derived for convective cooling or heating of liquids. Furthermore, it is shown how the local heat transfer coefficient can also be calculated without iteration in the corresponding cases of combined radiation and convection.

The calculation of heat transfer coefficients in heat exchangers usually requires an iterative technique if the heat transfer coefficients depend on the surface temperature of the wall. In the case of convective heat transfer this dependence is usually expressed by a wall viscosity correction factor

OILS

is first calculated using known correlations for the heat transfer coefficients. With this ratio A and eqn. (1) the true unknown wall surface temperature Tw is expressed by -l

(1)

with eye as the limiting value for zero driving temperature difference. Recently it was shown’ that, for the cooling of viscous oils in air-cooled heat exchangers where the air-side heat transfer coefficient is virtually independent of wall temperature, the correction factor K on the process fluid side can be calculated without iteration using Hack1 and Groll’s equation* K=G-HXtLX’

2. IMPROVED METHOD FOR COOLING VISCOUS

In the previous method’, which is described briefly in the following, the ratio A of the uncorrected heat transfer resistances,

1. INTRODUCTION

K = (Y/QI,

where T is the absolute temperature; T can be replaced by the difference T - 8 if a suitable value of 8 is known. in this paper an improved method is developed based on a’simplified correction equation for the wall viscosity. Furthermore, similar iteration-free procedures are developed for other cases where one of the heat transfer coefficients depends on the wall temperature.

AK&f&-

w

and with approximate AK*=-

(2)

T,$-

T’ W

The approximate

values K * and T$ this becomes (7)

value T.$ yields, according to eqn. (3),

where x = log (nT,/OT)

x* = Iog (rlT&/%T)

(3)

Applying eqn. (4) also for T, and r,$ and combining these three viscosity equations with eqns. (3) and (6)-(g) gives, as shown in ref. 1,

and G, W and L are empirical constants with two sets of numerical values for different ranges of X. The viscosity of the process fluid was assumed to obey Andrade’s law3 logqr=B+C/T

(8)

X -_= X*

(4) 233

AK*+T’/T AK •tT’/T

(91

W. ROETZEL

234

Solving eqn. (9) for X and introducing it into eqn. (2) finally leads to a cubic equation for K which can be solved in the usual manner. The resulting explicit equation for K is very complicated and inconvenient for quick calculations. However, the major disadvantage of the method’ is that one has to consider two sets of constants G, H and L. As the true wall temperature, and consequently X, are not known initially, K has to be calculated using one of the two sets. If the wrong range of X is obtained the calculation has to be repeated using the other set of constants. These disadvantages of the previous method can be overcome if the following new equation for the correction factor is used: y= 1

K-N

~

1

I-N

+MX

N=0.13

In cases where normal hquids are cooled or heated, the correction K is usually calculated according to (16)

where according to Hausens M=O.69

(111

Equation (10) yields the same accuracy as the original equation of Hack1 and Gro112and the simpler equation proposed by Hausen4. Intr~ucing eqn. (9) into the new eqn. (10) now yields the following quadratic form: K’+UK=V

3. HEATING C-JRCOOLlNG OF LIQUIDS

K = (~~/~~~)~

where for the total range OGXG4

peratures T and T’ the ratio A is calcdated according to eqn. (5). A wall temperature Ty, is assumed and the viscosity of the fluid under consideration is evaluated for this wall temperature and at the mean fluid temperature T. Then U, V and finally K are calculated according to eqns. (15) and (13). This procedure is much simpler than the previous one and is valid for the total range of X. As discussed in ref. 1, the method can be modified by replacing the absolute temperatures T and T’ in eqns. (15) by the temperature differences T - B and T’ - 8.

(121

or (13)

n = 0.14

(171

Substituting the variable X according to eqn. (3) K = exp (-nXln

10)

(18)

Using the arst term of a series expansion (see, for example, ref. 4) 1+z In -= 1-z

a

the exponential function of eqn. (18) can be approximated by li - 5Xin

IO-

K=

(14)

Substitntjng~ and M from eqn. (11) into eqn, (14) and similarly X* and K* from eqns. (8) and (7) yields the following expressions for U and V:

(201

The relative error of K according to eqn. (20) is less than 0.3% compared with eqns. (16) and (17) in its validity range 0.1 G nT/nT, g 10. Equation (20) can be rearranged in the form of eqn. (10) with the constants Nz-.1

M=p14 In IO

(21)

Substitution into eqn. (14) yields 1 l-T’/T

T’

ii T/Pw - 1 +AT

Thus the desired correction factor K can be calculated again without iteration. With the known tem-

-1

,__\

CALCULATION

OF HEAT TRANSFER

23.5

COEFFICIENT

and the correction can also be calculated using eqn. (13) for cases in which eqns. (16) and (17) are valid. Instead of eqns. (16) and (17), Gnielinski’ proposed the correction equation (23) With the realistic assumption that Pr also obeys Andrade’s law, one can derive the following equations in the same manner as for eqn. (22):

(24)

ference yields the heat transfer coefticient T4 - T4 c+=E;y-+=ET w

4. RADIATION

& = E(T4 - T$J

(25)

in which E is a (positive) constant, independent of wall temperature. This assumption is fulfilled with flame radiation in the total wall temperature range Ta T, > 0. In the case of selective gas radiation, eqn. (25) is exactly fulfilled only if T = T, > 0 or T > T, = 0. However, eqn. (25) may be a good approximation if T = T,, as may occur in heat exchangers, or if T% T,. Dividing ean. (25) bv the drivina temuerature dif\I<

(26)

CQ)=

lim

CY,= 4ET3

(27)

TWIT-‘1

This can be calculated without temperature. Using this limiting coefficient CY,can be calculated the correction factor Kr which (26) and (27):

r

knowledge of the wall value CQ, the true by multiplying by is the ratio of eqns.

=Lf~_~l-Kvlrr)~ 0~0-4

(28)

1 -T,fT

Dividing the numerator

by the denominator

yields

Kr=$[ 1 +$+($f +($)‘)

AND CONVECTION

In the previous cases of forced convection the dependence of the heat transfer coefficient ty on wall temperature is caused by the variation of the liquid viscosity with temperature. Even without this viscosity effect the heat transfer coefficient can still be strongly dependent on wall temperature if heat is transferred by radiation. This case is considered in the following where the liquid is now replaced by a gas or a flame which exchanges heat with the wall by radiation and convection. As in ref. 1, the convective heat transfer coefficient (Y, of the gas is regarded as being independent of wall temperature, as is the heat transfer coefficient a’ of the other fluid, It is assumed that the (positive) radiative heat flow per unit area from the gas or flame to the wall surface obeys the equation

“1

(Kv/04

1 - T,/T

As in the previous cases of forced convection, we now also define for ra~ation the l~iting value or0 of the heat transfer coeffcient a, according to eqn. (26):

K

Equation (24) must be used together with eqn. (13) if eqn. (23) is to be valid.

3 I-

for radiation:

_

.

(29)

Equations (28) and (29) show that Kr is a direct function of the ratio of the wall and gas temperatures only. If convective heat transfer also takes place, the heat transfer coefficient has to be calculated according to CY=iXrto, where a;. is considered to be independent perature. Using the limiting value

(30) of wall tem-

c&J= b,o + a,

(31)

and the correction factor K the total heat transfer coefficient according to eqn. (30) can be calculated from ~Y=(Y~K=~~-~;+cY,

(32)

which defines K in the same way as eqn. (1). Introducing into eqn. (32) the ratio $_.zK!L a;, +o,

(33)

and solving for K yields K=K,$+l-$ with Kr given by eqns. (28) or (29).

(34)

236

W.ROETZEL

Rearranging eqn. (6) one finds T, AK+T’/T -_= T AK+1

gives a quadratic

ctl-T&‘-9 d+tl -TwIT)

=l_

r

for K as in eqn. (12) with

(35)

where A is the ratio of uncorrected resistances defined by eqn. (5). Substituting T,/T from eqn. (35) into eqns. (28) or (29) and introducing either into eqn. (34) yields an equation in which K is the only unknown. However, this equation cannot be solved directly for K and so it is usually solved iteratively. An iteration-free solution is possible if, instead of eqns. (28) or (29) an approximate equation of the following form is used: K

equation

(36)

We first consider the case in which T,,,/T = 1, The constants c andd are determined so that for T,/T = 1 both the first and the second derivatives of the exact function eqn. (29) are equal to those of the approximate function eqn. (36). Differentiating eqn. (29) twice gives

(41) 0.8 < T,/T

< 1

Thus, after qo,,, czo and J/ have been determined according to eqns. (27), (31) and (33), and A has been determined according to eqn. (5), the values of U and V can be calculated from eqn. (4 l), and finally the required correction factor K from eqn. (13). Using eqn. (35) one can subsequently check whether T,/T lies in tne range where eqn. (36) is valid. With flame radiation, cases can arise where T,/T falls below 0.8 and eqn. (4 1) yields less accurate results. For the range 0.3 < T,/T < 1 .O the constants c and d in eqn. (36) should therefore be changed to d = 0.9774

c = 1.5616

(42)

This results in (37)

U=;

- 1 (43) 2.023

We then obtain from eqn. (36)

0.3 < T,/T< (38)

Combining

eqns. (37) and (38) yields (39)

In the range 0.8 < T,/T < 1.2 the relative error in K, according to eqns. (36) and (39) is less than 0.5%. Substituting for K, in eqn. (34) from eqns. (36) and (39) gives K

=

1

_

ti$(l - TwIT) :+l

Substituting

T,/T

-T,/T from eqn. (35) into eqn’(40)

- 1.023$+

1.598 $

1.0

The mean relative error of Kr according to eqns. (36) and (42) is around 0.7% with the maximum error at the point TWIT = 0.3 being - 1.6%. For the range 0 < T,/T < 0.3 another method can be derived assuming that (Tw/n4 = 0 in eqn. (28). Then 1

1

K’ = 4 1 - T,/T

(44)

with a maximum relative error of 0.82% for T,/T = 0.3. Substituting in eqn. (44) for T,/T from eqn. (35) and then introducing K, into eqn. (34) yields, after solving for K,

(40) finally

4(1 - T’/T) - A1//

(45)

CALCULATION

OF HEAT

TRANSFER

Thus the correction factor K for combined radiation and convection can also be calculated without iteration for small values of T,/T where the assumed eqn. (25) is also a good approximation for selective gas radiation. NOMENCLATURE

A

ratio of uncorrected heat transfer resistances, defined by eqn. (5)

B,C

constants dummy variables constant heat transfer area constants correction factor, defined by eqn. (1) constants exponent Prandtl number heat flux (per unit area and per unit time) absolute temperature of the fluid dummy variables

c,d E F

G,H,L K

M,N n Pr

4 T u, v,x,z

237

COEFFICIENT

Greek symbols heat transfer coefficient (dependent on a the wall temperature) wall thickness 6 dynamic viscosity of the liquid empirical reference value for the abso; lute temperature thermal conductivity of the wall material x ratio of uncorrected radiative and total 4J heat transfer coefficients, defined by eqn. (33)

Subscripts 0

C

m r W

uncorrected or limiting value for the zero driving temperature difference convection mean value for heat conduction radiation at the wall surface on the side where the heat transfer coefficient depends on the wall temperature

Superscripts * estimated value or determined with an estimated value cold air stream in air-cooled heat exchangers or fluid with a heat transfer coefficient independent of the wall temperature REFERENCES 1 W. Roetzel, Iteration-free calculation of the heat transfer coefficient in air-cooled cross-flow heat exchangers, Chem. Eng. J., 7 (1974) 79-81. 2 A. Hack1 and W. Groll, Zum Warmeiibergangsverhalten zahfltissiger idle, Verfuhrenstechnik fMuinz), 3 (1969) 141. 3 E. N. Da C. Andrade, Viscosity of Ii&ids, Nature (London], I25 (1930) 309,580. 4 H. Hausen, Verfahrenstechnik (Mainz). 3 (1969) 355,480. H. Hausen, Erweiterte Gleichung fur den Wlrmeiibergang in Rohren bei turbulenter Stromung, -. W&meStoffiiberfrug., 7 (1974) 222-225. W. Meyer zur Capellen, Intergraltafeln, Springer-Verlag, Berlin, Gottingen. Heidelberg, 1950. V. Gniclinski,Warmetlbergang bei der Stromung durch Rohre, in VDI-Wiirmeutlns, Berechnungsbtitter fiir den Wirmeiibergang, VDI-Verlag GmbH, Dusseldorf, 1974, Chap. 6b.