An insight into the contribution of the core friction loss, Afif, to the pressure drop, Afi, can be obtained if the heatmass transfer analogy is assumed applicable. In this case Seban’s analogy (77) is appropriate-i.e., 11 ~h~ ratio of h c l e a r ohannel was de-
- 47 --dGj.
characteristics Therefore,
of
units
(72).
L
APf clear
holenr ohannol
hpromoters
these
=
promoters
and
htnihulence promoters
termined in a study of the heat-transfer
I
The Fanning Equation can be used to calculate 4$, c l e a r , and Equation 3 to calculate A$. The respective ratios of 4fi,/Afi for units A, B, and C were 0.19> 0.65, and 0.62 (Re = lo4). The ratio for unit A indicates that the core pressure drop represents only a small part of the total pressure drop. Manufacture of this unit has been discontinued.
A. A. McKlLLOP and W. L. DUNKLEY University of California, Davis, Calif.
(Plate Heat Exchangers)
Heat Tra nsf er Turbulence promoters and flexible plate arrangements give high heat-transfer rates at low pressure drops MAIN OBJECTIVE of this study was to determine heat-transfer coefficients and temperature patterns in the three typical designs and to arrive at suitable correlations with which to predict these quantities for varied operating conditions. Plate heat exchangers lend themselves to mathematical analysis lvhen operating ccnditions approximate the idealizations that must be imposed. The experimental data provided the appropriate boundary conditions necessary for this study. These data were the entering and leaving fluid temperatures and the mass flow rates. The temperatures were measured by placing thermometer wells or thermocouples in the entrance dnd exit pipes of the sections. Mass flow rates were determined by volumetric timing. All tests on each unit were run with a constant hot-side fluid rate to simplify the analysis. Therefore, heat-transfer coefficient determinations were based on heating of a fluid. The temperature ranges covered were moderate (Figure 10). The study was limited in that water was used as both the hot and cold fluid and, therefore, the role of the Prandtl number could not be ascertained. Heat-transfer coefficients were calculated by two methods: the mathematical analysis reported here and the experimental correlation presented in McAdams (77). The latter predicts coefficients for clear-channel flow. Turbulent flow between flat plates has already been analyzed (74). I n plate heat exchangers the dynamics of fluid flow are essentially similar except that the turbulence promoters must be taken into account. I n effect, by promoting
E:?
740
turbulence, they cause complex velocity and temperature patterns that create a suppressed thermal boundary layer and thereby increase heat flow. The exact mechanism has not been determined, however. The idealizations required for analytical solution of the thermal characteristics in plate heat exchangers are: 1. That the over-all coefficient of heat transfer is constant throughout the length of the heat exchanger. This implies that all fluid properties are independent of temperature. 2. That the heat loss to the surroundings is negligible. This assumes that the two end plates serve as adiabatic walls. 3. That heat is not conducted in the direction of fluid flow by the walls or by the fluids themselves. 4. That the temperature and flow rate are uniform across the channel width.
Fortunately, the plate heat exchanger comes close to satisfying all these conditions. In operation. temperature differences between inlet and outlet are small. Thus, assumption of a constant over-all coefficient approxima tes the actual situation. The only surfaces exposed to the atmosphere, except the edges, are two end plates. Since each end plate is surrounded by dead air space and the opposite surfaces are nearly at the same temperature, the idealization of an adiabatic wall is a valid postulate. Condition 3 is also widely accepted for most fluids. The main concern, then, is uniformity of the temperature and velocity across the width (condition 4). One ex-
INDUSTRIAL AND ENGINEERING CHEMISTRY
ception to satisfaction of condition 4 is in the channeling effect around the edges of the plates. Although the short-circuiting fluid has a velocity higher and a temperature lower than those of the mainstream, the volume of fluid involved and The velocity and temperature differences are negligible in relation to heat transfer. Another exception is the diverging and converging of the flow channels at the ports. This condition applies only to a small portion of the heat-transfer area. IVith these idealizations it is possible to derive a set of linear differential cquations predicting the temperature profiles along the plates and, if sufficient boundary conditions are known, to calculate the heat-transfer coefficients. For any practical flow configuration, there are coo many simultaneous equations to solve conbeniently by any method other than a numerical procedure. Thus the IBM 704 high-speed digital computer was used for this investigation.
Theory of Analysis An application of the energy equation to the control volume shown in Figure 11 yields [U, - l ( t ,
- t, -
1)
+ U,(t,
- t, + 111 x b dz = cnwndt, (5)
This equation assumes that energy changes due to elevation difference and velocity head are negligible. The over-all heat-transfer coefficient for plate n-1 is defined by
PLATE HEAT EXCHANGERS in which h/ denotes the equivalent fouling factor on both sides of the plates. Furthermore, over moderate ranges of temperature and velocity, the heat-transfer coefficient can be correlated by
hD,
= Bn
(2)
k
I160
'
I
L
l
/
rn
(7)
Thus
because thermal properties are assumed to be constant, or
/hr.
If the cold fluid divides in previous or subsequent channels, moreover, the corresponding coefficient could be expressed in the same form as Equation 7a. I t is still possible, therefore, to express all heat transfer coefficients in terms of one unknown. On this basis, Equation G can be written as 1 = L(1 + uw-I hen
in which
This factor will be called the effective resistance. An equation similar to 8, written for plate n, defines the over-all heat-transfer coefficient as
65
in which
I
5
3 Channel
Unit 8
When expressions 8 and 9 are combined, UnVlcan be written in terms of
U,
Now Equation 5 can be expressed in terms of U, alone. T o generalize these equations, the foIlowing nondimensional terms should be substituted
All subsequent equations are dimensionless unless otherwise specified. Channel Unit C
Figure 10.
Temperature profiles for the cold-side fluid VOL. 52, NO. 9
0
SEPTEMBER 1960
741
, ~
The analytical solution for any type of plate arrangement, then, involves the solution of a system of n linear first-order differential equations in which n f 1 is the number of plates. The flow diagrams for the three units analyzed are shown in Drawing A. Applying Equation 11 to the three units gives the following sets of equations: Unit A dT1 - 3UbL (Ti dv WhCh
-
da
Plate n+l
7
7
(A-1 1
T?)
EL(TI+ T3 - 2T2)
dTz =
Plate n-2
h -
(A-2)
~ c c o
dTs = 3ubL(2T8 - T B- T,) dv WhCh
(A-3)
dT" = EL ( 2 T 4 - T3 dv
(A-4)
'rj)
~ e c c
3UbL = (2T5
dT5
da
WhGh
-
T4
UbL c6 [(T5 - To) r' da : =
~
WeCc
- TF,) (A-5)
/
@3(Ti - Ta)] (A-6)
dTc - 2UbLOs
-dv- _ _ _ Whch
dT, -
UbL9a
da
WCCC
(To
+ T8 - 2T;)
(A-7)
(2T8 - T7 - Tg)
(A-8)
dTg -_2UbL@3 _ (Ts - TB) da
Figure 1 1
4 wnti
Control volume was used to derive the energy equation
e
The following boundary conditions apply :
in which
(A-9)
WhCh
Ti=T3 T .0 = rr7 T, = given TB T1 =
a = O
in which
+ hcR + 9-'1> (1 f h,R + I) 42
(1 ~~
(2)"
@ 1 =
=
9 2
$4
=
($)"L Ply: n = O
TP = T8 = given T7 = Tg = 1.0 2
=
T4
'rF,=
T~
l'nit B dTi - -_2UbL@6 _ (TI - T2) da wccc
dT2 - lJbL$s -- (TI ~
d?l
dT3
WhCh
-
d7 dT4 dV
2UbL96 WCCO
+ T3 - 2T2)
(2T3 - T? - T4)
- lJbL .[ $ s ( T ~- T1) (T5
-___
wccc
("4 4- Ts
dTs - UbL (T5 dv WhCh
-
-
(B-2) (B-3)
T4)l (B-4)
2T5)
(B-5)
+ Ti - 2Ts)
(B-6)
T8)
wccc
dTs = UbL (T7 - T8) d7 WhCh 742
(B-I)
+
dT,= ubL(2T, - 're dq
=
2 -%,,
a = l
TI
+2 T3 =
T5
T B = Tg = TF, = T8 = 1.0
T g
=
TR = Tlo = 1.0
TI = given T3 = T j T, = T,
These equations are limited to the idealizations stated earlier. The solutions of units A and B require that initial values be assumed for h, and m. But the h,R term is small compared to the
+ @ term:
and a reasonable estimate
should not materially affect the value of 4'3 or $6. Gill's variation on Kutta's fourth-order approximation was used ( 4 ) in solving these differential equations. Gill's program. hoivever, assumes that all the initial conditions are known. Since some of the conditions were known only at the final boundary, the plate heat-exchanger investigation required a modification of Gill's program. The rnathematical procedures of this program are outlined elsewhere (12).
WhCh
dTs - UbL da
'" h,,
1
+ Tg
?'z
=
1
T-i-T 1 = T3 = T5 = _
1
rl =
+s
The following boundary conditions ap-
'The following boundary conditons apply: 7 = 0
('2) (2)
(B-7)
(B-8 1
INDUSTRIAL AND ENGINEERING CHEMISTRY
Resuits and Discussion
The numerical solutions of the differential equations defining the energy balances of the three heat exchangers involved a number of computer runs for each flow rate, The reason was that the solutions depended upon the value of the exponent rn assumed in determing the @ function. The equations for units A and B contained this q5 factor explicitly
PLATE HEAT EXCHANGERS because the ratio of the fluid flow rates was not constant for all flow channels. Initially for units A and B, numerical solutions were made a t selected intervals of 0.05 in the value of m in the range 0.6 to 0.8 for each flow rate. The first four runs on each unit showed that a change of the exponent by 0.05 did not significantly alter the value of the over-all coefficient (less than lyo). Since this change was so small, subsequent runs were calculated using only one exponent. This value, based on the first four runs for each unit, was 0.7 for A and 0.8 for B. Therefore, in the analysis of the data it was assumed that the over-all heattransfer coefficient was independent of rn. This does not mean, however, that the individual heat-transfer coefficients are independent of m. Over-all Heat-Transfer Coefficients. The only variable for a specific value of rn in the determination of the over-all heat-transfer coefficient was the coldside heat-transfer coefficient. Thus, from 1 Equation 6 the total resistance, - can
U'
be expressed as the sum of a constant resistance and a variable resistance. 1 Since the variable resistance is -, which hc from Equation 7 is proportional to
fer coefficients for unit A were calculated by Equation 9 with an R of 0,00045. The method of least squares showed that the exponent was 0.7. Rohsenow (76),who correlated experimental data for flow in many channel configurations, noted that all results fall between the exponent values of 0.6 and 0.8, depending on the shape of the flow path. For units A , B , and C, the exponents derived from Equation 12 agree with Rohsenow's findings. Unit B approximates Rohsenow's conclusion that flow along regular channels which uniformly converge and diverge corresponds closely to an exponent of 0.8. Unit C illustrates his conclusion that irregular channels-ones that diverge and converge in all directions-yield an exponent value of around 0.6. Unit A illustrates semiuniform flow and, as expected, is sufficiently irregular to correlate a t a n exponent considerably below 0.8. Troupe (20) confirmed these findings. Individual Heat Transfer Coefficients. Once the proper exponents were found by correlating the over-all heat-transfer data, the individual coefficients for units B and C were easily
determined, because they are the reC, ciprocals of the - term. The indiw2
vidual coefficients for unit A were calculated on the basis of the assumption discussed above. These coefficients were then plotted in the conventional manner of Nusselt number us. Reynolds number. The equivalent diameter in the Nusselt number was based on measurements made at 70' F. T h e thermal properties of the fluid were evaluated at a bulk mean temperature of 1 0 5 O F. The data for unit A , therefore, were recalculated for this new temperature (Figure 13). T o gain further insight into the meaning of the analytical results, another method was employed for calculating heat transfer coefficients. McAdams ( 7 7 ) states that for turbulent flow of water in rectangular passages, with Reynolds numbers ranging from 3500 to 27,000 and aspect ratios of 1 to 7.9, the data correlated well when based on the following equation :
1
-
W*'
the total resistance can be written as follows:
in which a, is constant and equal io the sum of the hot-side resistance, the plate resistance, and the fouling resistance. The reciprocal of the last term represents the product or cold-side heattransfer coefficient. Furthermore, the
Figure 12. Regression line for the total resistance to heat flow as a function of flow rate, based on data calculated for the best value of m
Meon fluid temperature 155OF
~
I 1
C,
hot-side coefficient is given by - since WIE"
both fluids are water. Because plate resistance is easily determined, the fouling factor, h,, can be evaluated. Applying the least squares method to the data represented by Equation 12 for various values of m gave the best value of the exponent for correlation of the individual heat-transfer coefficient according to Equation 7 . For units A , B , and C this procedure yielded respective answers of 0.7, 0.8, and 0.67. Figure 12 shows the results of these analyses. The accurate value of the intercept constant, a,, depends on experimental runs over a wide range of flow rates. The equipment available limited this flow range, especially for unit A . For the above analysis on unit A , R turned out to be negative which is impossible (Figure 12). Since for Units B and C the effective resistances were found to be 0.00052 and 0.00039, respectively, the final values of the individual heat-trans-
I
I
I
2
I
3
4
lo3 UnitA
R,= 00052 Meon fluid temperature
R,= 000393
105'F
Mean fluid temperature 105*F
a5
10
---&--A &x,
-$xt,3
Unit B
io3
,
Unit C
VOL. 52, NO. 9
0
SEPTEMBER 1960
743
1
this factor, however, is small for most correlations. I t thus seems reasonable to use a value of the exponent of 1/3 for estimating heat-transfer coefficients for other fluids. Caution must be exercised, however, when making such an extrapolation for food products, since many are non-Newtonian fluids for which the proper function of the viscosity in pressure drop and heat transfer has not been determined. More investigation is needed even before direct heat transfer correlations can be made on such fluids.
70
Z
= concentration,
t
7
= height, ft. = dimensionless distance
0 0,
= time from first measured:par-
= time, sec.
ticle, sec. 1.1 p
= viscosity, Ib.,/hr.-ft.
T
= dimensionless temperature = notation for function defined in
= density, Ib.,/cu.-ft.
q5
paper
Subscripts app = apparent = cold-side fluid h = hot-side fluid j = general term n = general term p = plate 1, 2 = channel number c
Acknowledgment Meon fluid tempetalute 1 O P F
, 2
,
I
4
, ! , , I 8 IO4
6
Re=
(p)
2
3
Figure 13. Nusselt vs. Reynolds' number evaluated from (C/w)"" in Figure
12 Further increase of the aspect ratio appears to have little effect (9). The result of this calculation is shown in Figure 13. T h e importance of this method is that it forms a lower limit of expected heat transfer coefficients and thus helps evaluate the role of the turbulence promoters. The data correlated for moderate temperature differences and Reynolds numbers according to the dimensionless equation :
hA = B,(Re)m k I t did not seem advisable, however, to extrapolate the data beyond the Reynolds numbers shown in Figure 13. This restriction is particularly true for those lower values of the Reynolds number for which other exponents might be more appropriate. In addition, the effect of the turbulence promoter design on the exponent has not been studied enough to make extrapolation possible for other designs. The increased heat transfer attributable to turbulence promoters can be estimated by comparing the three analytical curves with the bottom curve in Figure 13. The relative effect of the different turbulence promoters is indicated by a comparison of analytical curves for a given Reynolds number. The analytical solutions for units B and C yield heat-transfer coefficients that are 6070 greater than for unit A . Temperature Profiles. Typical temperature profiles for specified flow rates in the three units are shown in Figure 10. These profiles are drawn from the temperatures calculated from the analytical solution. Prandtl Number Effect. This study did not evaluate the role of the Prandtl number. The range in the exponent of
744
The authors are indebted to the M.I.T. Computation Center for use of the IBM 704 for carrying out the major part of the heat-transfer analysis. Special thanks are due Marion Callaghan for assistance in this phase of the work. The authors thank W. M. Rohsenow and J. B. Powers for guidance and encouragement. G. W. Putnam, Creamery Package Mfg. Co. ; James McCarty, Milk Stop; and F. L. Morris, Sanitary Dairy, generously provided plate heat exchangers for the study.
Nomenclature A A,
= area, sq. ft. = channel cross-sectional area, sq.
2b
=
=
g gc
= = = = = = = =
H
=
h
=
h,,
=
k
=
L
=
m n
=
C c
De
f G
p
= = =
ft . constant constant channel width, ft. constant heat capacity, B.t.u./lb.,, " F. equivalent diameter, ft. friction factor mass velocity, lb.,,/hr.-sq. ft. acceleration of gravity, ft./sec.2 dimensional constant, Ib.,-ft./Ib.f-sec.2 pressure, inches of flowing fluid individual heat-transfer coefficient, B.t.u./hr.-sq. ft.-" I?. fouling factor, B.t.u./hr.-sq. ft." F. thermal conductivity, B.t.u./hr.ft.-" F. channel length, ft. exponent exponent pressure corrected for elevation difference, Ib.,/sq. ft. core friction pressure, lb.,/sq. ft.
R
1 = effective resistance, -
Re
=-=-
~h
= = = =
t
to
Lr
w x xo
INDUSTRIAL AND ENGINEERING CHEMISTRY
4w rh
= = =
hf
+ 2x
2w ASP & hydraulic radius, ft. temperature, " F. inlet hot-fluid temperature, " F. over-all heat-transfer coefficient, B.t.u./hr.-sq. ft.-" F. mass flow rate, lb./hr. thickness, ft. channel thickness. ft.
References (1) Allen, C. M., Taylor, E. A., Trans. Am. Soc. Mech. Engrs. 45, 285-341
(1923). (2) Cuttell, J. R., "The H.T.S.T. Plant. An Introduction to Technique, Control and Management," Dairy Industries, Ltd., London, 1948. (3) Fay, A. C., Fraser, J., J . M i l k Technol. 6, 321-30 (1943). (4) Gill, S., Proceedings of the Cambridge Philosophical Society 47, 96-108 (195 1 ) . (5) Goodman, H. F., Proc. 73th Intern. Dairy Congr. 3,772-4 (1953). (6) Hansen, S. A., Wood, F. W., Thornton, H. R., Can. J . Technol. 31, 231-9 (1953). (7) Jordan, W. K., Holland, R. F., White, J. C., J . Milk and Food Technol. 15, 155-8 (1952). (8) J . Milk and Food Technol. 13. 261-5 (1950). 19) Kavs, W. M.. London. A. L . . "Compact 'Heat Exchangers," The National Press, Palo Alto, Calif., 1955. (10) Lawry, F. J., Chem. Eng. 6 6 , 84-9 (1959). (11) McAdams, W. H., "Heat Transmission," 3rd ed., McGraw-Hill, New York, 1954. (12) McKillop, A. A , , M.E. thesis, M.I.T., 1959. (13) Murdock, D. I., Brokaw, C. H., Allen, W. E., Food Technol. 9 , 187-9 (1955). (14) Pai, S., "Viscous Flow Theory IITurbulent Flow," Van Nostrand, Princeton, N. J., 1957. (15) Prandtl, L., "Essentials of Fluid Dynamics," Blackie and Son, Ltd., London and Glasgow, 1952. (16) Rohsenow, W. M., private communication, M.I.T. (17) Seban, R. A., Trans. Am. Sac. Mech. Engrs. 72, 789-95 (1950). (18) Stewart, G. N., Am. J . Physiol. 57, 27-50 (1921). (19) ,Troupe, R. A , , Morgan, J. C . , Prifti, J., Chem. Eng. Progr. 56, 124-6 (1960). (20) ,Troupe, R. A , , Morgan, J. C., Prifti, J., unpublished data. (21) Walzholz, G., Milchw. Forsch. 20, 259-78 (1940). (22) White, H. L., A m . J . Physiol. 151, 45-57 (1947). ~I
RECEIVED for review November 20, 1959 ACCEPTED May 16, 1960 Work supported in part by funds from the California Dairy Industry Advisory Board.
