I
WILLIAM SCHOTTE Engineering Research Laboratory, Engineering Department, E. I. du Pont de Nernours & Co., Inc. Wilrnington, Del.
Heat Transfer to a Gas-Phase Chemical Reaction This is a subiect of growing importance,, which has recently received wide attention from aerodynamists in the field of hypersonic flow
IN
RECENT YEARS, renewed interest has been shown in heat transfer to gasphase chemical reactions. Altman and Wise (7) mention that heat transfer from flames to cooled surfaces can be expected to show an increase because of the heat release resulting from the recombination of dissociated species in the boundary layer and on the surface. Ziebland (76) has discussed heat transfer problems in rocket motors. He reports that experimental values may exceed by a factor of 3 the computed figures of convective heat exchange between the hot combustion gases and the cooled walls. The experimental study described here gives another example of the effect of a gas-phase reaction on the heat-transfer rate. The theory will be limited to cases where the reaction rate is not controlling. I t is assumed that the reaction rate is so rapid that chemical equilibrium is attained throughout the reactor. Although some of the theoretical work is old, only in the last few years have correct relationships been proposed. I n 1904, Nernst (72) derived an equation for the thermal conductivity of a dissociating gas. Even though this equation is theoretically in error, the few experimental measurements made up to 1951 have been claimed to agree with Nernst's theory. This may have been the result of poor experimental technique and/or large errors in the estimated
physical properties of the gas under study. Prigogine and Buess (73) and Hirschfelder (7) have recently derived expressions for the thermal conductivity of a dissociating gas, which are theoretically correct. Their theories have not been compared with experimental data, as no new experimental work has been
reported and the older information is unreliable. Since 1954, in a few articles the thermal conductivity theory has been extended to cover convection heattransfer problems. Although these investigations have made important contributions, they are not directly applicable to the solution of practical problems. Design methods should be developed and experimental work is needed to check the theory.
Theory of Heat Transfer in laminar Flow The method of Graetz has been used to derive an expression for the heat-transfer coefficient in laminar
flow through a tube with the reaction:
A
+ 2B
A good description of the Graetz solution for heat transfer without reaction is given by Drew ( 4 ) ,whose derivation has been followed here. Consider a tube of radius r. The
diffusion equation for the binary reaction mixture is :
The molal mass balance for component A due to diffusion and chemical reaction is in differential form :
dY.4, 0)
where the function +CyA, 0) is the loss of A due to chemical reaction. Since bmyA/bz = mb In (1 yA)/bz, we have :
+
VOL. 50, NO. 4
APRIL 1958
683
Equation 10 indicates that the appropriate average value of k’ is a distance-integrated average. The right-hand side of Equation 10 must equal the heat input to the gas stream :
where But m = pu.
Then:
D’Gc,’ ( t ~ tl) =
I t will be shown later that: D A Bp(AH)’ 2R T 2
A similar heat balance gives :
CY
(1
td(1 -
- CY) +I=-
k
k’ k’
the ratio of the “effective” thermal conductivity to the normal value. Similarly,
The function 4 ( y A , 0) can be eliminated from Equations 1 and 2 :
(3)
(AH)’ CY ( 1 - a ) 2R T z @,..
I
+1
CP, CP
the ratio of the “effective” to the normal specific heat. Then
Equation 7 is the same differential equation as was used by Graetz and Drew, except for the additional factor,
Substitution of Equations 5 and 8 gives : t z - 61
(t,
- t i ) ( l - 8P2)
The arithmetic-average temperature difference, on which the heattransfer coefficient in laminar flow is based, will be :
Then :
21 h(t, -
ti)(l
+ 8Pp)rDL =
E.
Consider a parabolic velocity profile: u =
2
b
- (1
- p)
Y
For the experimental N204-SO2 system, it was found that E is close to unity and does not vary much with concentration. The temperature distribution is then nearly the same as for heat transfer without chemical reaction, It is believed that this is also the case for other systems. The solution of Equation 7 will be similar to that found by Graetz and Drew:
Substitution of the expressions for
$ and e (Equations 5 and 8) gives:
or
p
;(&)(-;)
1 - 8P
hD =
(11)
where PZ = 0.1024e-7.314+eL 0.0122~-44.61~~~. . . . . . Here
+
where 4 is a complex function of $6 z and s, the exact nature of which can be found in Drew’s paper, and t l is the inlet temperature of the fluid. The rate of heat transfer is given by:
Call
then Equation 4 reduces to : AH b In (1
+y
a2
~ -) 2k‘
D ( t * - I,)
a
1
- t,
bS
Using chemical equilibrium relationships, it is found that: dt
+YA)
= -oi
(1 -
where PI = 0.7488e-7.314#e 0.544ed44.61*62 . , . . . . .
+
AH 2RT2 ~
+
The total rate of heat transfer over the tube length, L: is:
Substitution into Equation 6 gives the approximate equation :
(1 - 8P2) (10)
or
684
where PZ = 0.1024e-7.314#i‘L 0.0122~-44.61ie~ . . . . , ,
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
k‘L *eL = r- 2 ZcJC*I
It is seen that hD/k’ is a function of
wc,’/k’L.Graphically, the relationship is shown in Figure 3 as the curve for heat transfer without natural convection. I t is the same relationship as that for heat transfer without chemical reaction, except for the use of the effective physical properties in place of the thermal conductivity and the specific heat. The above derivation has shown that, at least for laminar flow in a circular tube, the conventional heattransfer correlation can be used by employing an effective thermal conductivity and an effective specific heat. The appropriate effective thermal conductivity is a distanceintegrated value. Temperature distribution is only slightly affected by the chemical reaction.
(F)
which can be shown to give :
dln(1
+
+
General Theory The theory of heat transfer in laminar flow through a tube has been derived for
HEAT T R A N S F E R TO G A S - P H A S E R E A C T I O N a simple dissociation reaction: A + 2B. Although this is the most elementary case of practical interest, it is used here as the basis for the correlations applicable to other cases. The derivation has shown that to a good approximation, the conventional heat-transfer correlations for flow without a chemical reaction can be used for dissociating gases by employing “effective” thermal conductivities and “effective” specific heats. For a better understanding of these effective physical properties, derivations are given below for the reaction: A F! B C, where components B and C are present in equal molal concentrations. The effective specific heat is defined as the number of British thermal units required to raise the temperature of 1 pound of material 1” F. Although equations can be written for point values of the specific heat, it is simpler and more useful to determine an expression for the average value over a specified temperature range. Using the definition, it is found that
+
cp’(t2 - t l ) = Z[(H,n,)t, -
where the heat of dissociation, AH = HB Hc - HA, has been substituted. The general equation for multicomponent diffusion is:
+
which for our three-component system reduces to :
for the case where yB = yc. This equation can be simplified by introducing a mean diffusivity, 8:
Since Equation 17 is not explicit in terms of mi,Hirschfelder has suggested the use of the equation:
where the multicomponent diffusion coefficients, D,, ’, have been introduced. These multicomponent diffusion coefficients can be calculated from the binary diffusion coefficients and the composition of the mixture by rather involved equations of Hirschfelder, Curtis, and Bird ( 8 ) . Substitution of Equation 24, into Equation 23 gives the general expression : k’ = k
-2 ~ H y i,j
*
z D ~ , ’ M ~ (25) dt
Special solutions of Equation 25 should be calculated for each case. Even for a three-component system, such as A + B C, where B and C are not present in equal molal concentrations, the solution is rather complex. Our derivation has shown that:
+
We then get:
k‘=k+
HA)^,]
2
where n, is the number of pound moles of component i per pound of mixture. In more usual notation:
or m A =
tz
- tl t
k’=k-
For the three-component case, A C, where
+
YA =
YB:
Calculation of the effective thermal conductivity is more difficult. It is best to start with the defining equation for the effective thermal conductivity:
Separation into heat conduction and diffusion terms gives:
HBmB
+ Hcmc
(15)
where m is the molal rate of diffusion and H is the enthalpy carried per mole. The relationship for the effective thermal conductivity is found from Equations 14 and 15: k’
k - (HAm.4 f HBmB
pB(AH)
dt
For nearly instantaneous reactions, the chemical equilibrium relationships can be used. According to thermodynamics: d l n K - AH dT RTZ where the equilibrium constant, K , is given by :
Substitution of Equation 20 into Equation 19 gives after a rearrangement:
or in terms of the fractional degree of dissociation, a :
Combining Equations 18 and 21 gives the final expression :
dx
f Hcmc) 2
Since in steady-state diffusion mB = mc = - m A : k‘ = k
YA)
which can be substituted into Equation 16 to give: * dln(1 +YA) (18)
(S) ,] M
- d In (1dx+
-PD
+ ma(AH) dxdt
_-
(16)
More general equations for any multicomponent mixture can be derived as shown by Hirschfelder (7).
To illustrate the importance of the diffusional contribution to the effective thermal conductivity, calculations were made of the variation of k‘ and k with temperature for the dissociation of nitrogen tetroxide to nitrogen dioxide. A conventional correlation was used for k, and k’was calculated from Equation 22. The results are shown in Figure 1. The temperature range is from 22” C., the normal boiling point, where CY = 0.165, to 12OoC., where a = 0.950. It can be seen that the effective thermal conductivity goes through a maximum a t about 54” C., where k’ = 12.3 k. This means that heat conduction is twelve times as rapid as for a nondissociation system. Similar results will be obtained for the effective specific heat. For successful correlation of convection heat-transfer coefficients, appropriate average values of the effective thermal conductivity and specific heat should be used. In turbulent flow, the physical properties are properly evaluated a t the film temperature, which is approximated by the arithmetic average between the wall temperature and the bulk temperature of the gas. While the film temperature is adequate for evaluation of such properties as density, viscosity, conventional specific heat, and conventional thermal conductivity, it is not a good approximation for effective thermal conductivity and effective specific heat. As c,,’ and k‘ will go through a maximum when plotted against temperature, integrated-average values over the temperature range from the bulk VOL. 50, NO. 4
APRIL 1958
685
.1
.09
JL
c: .07
2
.06
CY
(1
- a)&
(29)
The theory for heat transfer in laminar flow has shown that the temperature profile is only slightly affected by the dissociation reaction. The change in the bulk temperature with distance can then be expressed by :
-08
3U
pw
e
+’
where t l is the inlet Differentiation gives :
>
Y
temperature.
?
L
.OS
s t L
Substitution into Equation 29 gives:
n .04
8
a ( 1 - CY)(t T2
JT;
03
t1)”2
dT
p?
W
I
Substitution of Equations 27 and 30 yields finally :
c
.02
.a1
t ~ ) ”d ~In (1
0
40
20
60
80
100
M
TEMPERATURE, “C. Figure 1,
Thermal conductivity of dissociation of nitrogen tetroxide
temperature to the wall temperature will be more accurate. Equations 12 and 13 give such an average for the effective specific heat. Integration of Equation 22 gives the average effective thermal conductivity for a three-component system with equal molal product concentrations: kj’ =
kj
+1 tw - t b
Tw P J ’ b f ( A H ) 2a (1 - a)dT 2RT2
=
kf
+
S T b
2R a (1 - a ) d
(- +)
(26)
Equation 1 9 can be used to find the expression : d
( _ _k)
=-
FHddnK
When K is expressed in terms of the fractional degree of dissociation, CY,we obtain:
686
2R
da
( $) _- -AH a ( I -
d - -
CY~)
Substitution of this expression in Equation 26 leads to the final equation:
For laminar flow, the bulk temperature of the gas is used to evaluate the physical properties. The heat-transfer coefficient varies with distance from the entrance, and average values over the length of the equipment, such as a tube, should be used. Again, Equations 12 and 13 serve to calculate the average effective specific heat over the temperature range from inlet to outlet bulk temperatures. As the heat-transfer coefficient varies with distance along the length of the tube, a distance-integrated average value is used in the heattransfer calculations. The appropriate value of k’ should be an average which has been integrated over the length of the tube:
INDUSTRIAL AND ENGINEERING CHEMISTRY
+
CY)
(31)
where t~ is the outlet gas temperature. Unfortunately, a graphical integration is required to solve Equation 31. In the experimental investigation, described below, it was found that only a small error was introduced by using a temperature-integrated average effective thermal conductivity in place of Equation 31. The inlet and outlet bulk temperatures were used in Equation 28. While Equation 31 is recommended for accurate work, first-order approximations can be obtained from Equation 28. Calculation of the effective thermal conductivity of complex reaction mixtures is a time-consuming procedure. Simple analytical solutions are not obtained. A numerical calculation is required to obtain the appropriate average values for turbulent and laminar flow. For a tentative design, an approximate value of the average effective thermal conductivity may be computed from the average effective specific heat. Work with the nitrogen tetroxide-. nitrogen dioxide system showed that (k’/k)(c,,ci) was not too far from unity over the complete concentration range. It can be shown that the relationship k‘ = k
(5)
will hold exactly when the molecular diffusivity is equal to the thermal diffusivity. For a pure gas this is a good assumption, but for a mixture the equality is only a rough approximation.
H E A T TRANSFER TO G A S - P H A S E R E A C T I O N
To VENT
T
MIXING PRESSURE
DE-ENTRAINMENT SECTION
NDENSATE BLEED SiEAM Figure 2.
Many safety precautions were incorporated in the experimental equipment
Experimental Study T h e dissociation of nitrogen tetroxide to nitrogen dioxide was found to be a convenient reaction for study. Accurate measurements can easily be made, as the conditions are close to atmospheric pressure and room temperature. T h e reaction is not catalyzed by the tube wall. Because of the extremely high reaction rate, chemical equilibrium may be assumed a t any instant. The heat of dissociation is large enough to give a manyfold increase in the rate of heat transfer. Finally, a simple reaction: Nz04 $2N02 takes place, so that calculations are made easier. A sketch of the experimental equipment is shown in Figure 2. Liquid nitrogen peroxide was vaporized in a boiler by means of a 1-kw. Calrod heater. Droplets were removed in a de-entrainment section, consisting of a baffle and two screens. The gas flow rate was determined by a calibrated rotameter. A 2-foot long calming section was used to stabilize the flow pattern. The gas passed from a heattransfer section 3 feet long to a mixing section 9 inches long. The mixing section was provided with close-fitting, rolled-up and twisted screens to give the true bulk
outlet temperatures. Finally, the gas was cooled and condensed in a 21-foot condenser for return to the boiler. Atmospheric pressure was maintained by a vent line. A single, stainless steel tube 0.443 inch in inside diameter and 0.028 inch in wall thickness was used for the calming, heat-transfer, and mixing sections. Heat conduction along the length of the tube could be neglected. Heat transfer in the calming section was made negligible by adjusting the room temperature to within 2' C. of the inlet temperature and by covering the tube with aluminum foil to minimize radiation. The mixing section was insulated and then covered with a heating tape. The heating current was adjusted so that the temperature of the insulation was equal to the outlet temperature. Steam heating was used in the heating section. Water was removed from the building steam by bleeding off condensate and subsequent superheating. The pressure was regulated and measured. The heating current in the superheater was adjusted so that the measured steam temperature was 5' to 10' C. above the saturation temperature. Heat losses from the steam jacket were minimized by the flow of steam through a second steam jacket. Condensate from the inner
jacket filled a cooler and part of a connecting glass tube. T o obtain a heat balance, the condensation rate was determined by keeping a constant level in the glass tube and by weighing the collected water. The following measurments were made for each run: Gas flow rate, inlet pressure and temperature, and outlet temperature. The pressure was measured with a mercury manometer. For temperature measurements, calibrated thermometers were used, giving readings to better than 0.l0 c. Condensate flow rate, and steam inlet pressure and temperature. Barometric pressure and room temperature.
In calculating the experimental values of the heat-transfer coefficients for air and for laminar flow of nitrogen peroxide, the inside wall temperature was assumed to be equal to the saturation temperature of the steam. The high heat-transfer coefficients for turbulent flow of nitrogen peroxide required that a small correction be applied to account for the steam-side and wall resistances. Many safety provisions were necessary for handling the toxic and corrosive nitrogen oxides. All metal in contact VOL. 50, NO. 4
APRIL 1958
687
Table I.
Data and Calculated Results for Nitrogen Peroxide in Turbulent Flow Run 1
Flow rate, lb./hr. Inlet temperature, C. Outlet temperature, C. Degree of dissociation at inlet Degree of dissociation at outlet Rate of heat transfer, B.t.u./hr. Steam saturation temperature, C. Rate of steam condensation, lb./hr. Heat release from steam, B.t.u./hr. Log mean temperature difference, F. Over-all U , B.t.u./(hr.) (sq. ft.) (" F.) Measured h, B.t.u./(hr.)(sq. ft.)(" F.) For inlet conditions Film temperature, C. KRe
= DVbYf/Ff
cpf',
B.t.u./(lb.) (" F.)
kf', B.t.u./(hr.) (sq. ft.) (" F./ft.)
(%)f
Predicted h, B.t.u./(hr.) (sq. ft.) (" F.) For outlet conditions Film temperature, C. . v =~ DVbrf/pf ~ c p / , B.t.u./(lb.) (" F.) k j ' , B.t.u./(hr.)(sq. f t . ) ( O F./ft.)
($), Predicted h, B.t.u./(hr.)(sq. ft.)(O F.) Av. predicted h, B.t.u./(hr.)(sq. ft.) ("
F.)
Run 2
Run 3
Run 4
16.62 23.22 66.86 0.165 0.600 2228 101.9 2.513 2460 97.2 65.8 68.5
9.83 22.63 69.92 0.167 0.642 1447 101.9 1.627 1590 93.7 44.3 45.5
15.53 23.31 70.84 0.168 0.645 2264 113.9 2.644 2550 114.9
13.02 23.30 68.36 0.168 0.623 1823 101.9 2.072 2025 95.3 55.0 56.9
62.56 11,180 1,589 0.0776
62.26 6,620 1,583 0.0771
68 60 9,960 1.473 0.0717
62.60 8,770 1.588
0.697
0.696
0.718
0.697
69.2
44.9
58.7
56.6
84.2 13,280 1.443 0.0645
85.9 7,970 1.366 0.06105
92.2 12,060 1.217 0,0539
85.2 10,430 1.407 0.0618
0.830
0.834
0.860
0.847
71.2
44.6
55.3
56.4
70.2
44.7
57.0
56.5
Mith nitrogen peroxide was stainless steel. Teflon tetrafluoroethylene resin and African Blue asbestos were used as gaskets and packing rings. The system was kept anhydrous by the presence of phosphorus pentoxide in the boiler and Drierite desiccant in the vent line. Many provisions were made to ensure adequate safety in case of equipment failure. The equipment was tested with air flow. During the runs rvith air, the boiler was closed off from the rest of the system and air was introduced as shown in Figure 2. The initial runs with air indicated that a few improvements in temperature measurement were neces-
56.7 58.8 I
0.0774
sary to obtain accurate data. After the revision of the equipment, one run was made in the turbulent flow range and a second one in the laminar flow range. For turbulent flow. the heat-transfer coefficient was predicted from the standard correlation :
which includes the corrected coefficient of 0.021, as suggested by hlcAdams ( 7 0 ) . The calculated heat-transfer coefficient was 17.9 B.t.u./(hr.)(sq. ft.) (' F.),which compares very well with the measured value of 17.6 B.t.u./(hr.)(sq. ft.)(' F.). The laminar flow run gave a measured
value of 1.63 B.t.u./(hr.)(sq. ft.)(" F,), the same as predicted from the graphical relationship . given by McAdams- (70) for a parabolic velocity profile. Four runs with nitrogen peroxide in the turbulent flow range covered a flow range of :trRe = 12,000 to :Va, = 20,000, expressed as the Reynolds number based on inlet bulk flow conditions (.\k, = DVbyb/fiLg).The saturation temperature of the steam was varied from 102' to 114" C. Measured overall heat-transfer coefficients were corrected for the small steam-side and wall resistances. Asteam-side heat-transfer coefficient of 3000 B.t.u./(hr.)(sq. ft.)( o F.) was assumed. 'The calculated results are shown in Table I. The measured heat-transfer coefficients were based on the rates ol heat transfer to the gas stream: as calculated from its temperature rise. Table I shows that the rate of heat transfer calculated from the condensation rate of the steam is about 12% higher. An analysis of these data and those in the laminar flow range seems to indicate that the "superheated" steam still contained some moisture. Some of the discrepancy is, of course, due to heat losses from the steam-side. Five runs were made in the laminar flow range. The flow rate, expressed as the Reynolds number based on inlet, bulk conditions, varied from 11;1, = 1100 to A,, = 2100. Tube wall temperatures ranged from 102" to 120" C. Table 11 shows the calculated results. Here, the heat release by the steam was 4 to 20y0 higher than the heat input LO the gas. Part of the difference appears to be caused by wet steam. For both laminar and turbulent flow, the measured heat-transfer coefficients were six to eight times as high as would have been predicted for heat transfer without a chemical reaction. Correlation of Results
Table 11.
Data and Calculated Results for Nitrogen Peroxide in Laminar Flow
Flow rate, Ib./hr. Inlet temperature, O C. Outlet temperature, O C. Degree of dissociation At inlet At outlet Rate of heat transfer, B.t.u./hr. Steam saturation temperature, O C. Rate of steam condensation, lb./hr. Heat release from steam, B.t.u./hr. Arithmetic mean temp. difference, O F . Measured h, B.t.u,/(hr.)(sq.ft.)(O F.)
Lz
( t - tl)'"%ln (1
+ a)
Run 1 1.763 22.51 84.71
Run 2
Run 3
Run4
Run5
1.363 23.30 88.26
0.891 23.37 93.53
1.765 23.08 94.33
1.363 22.85 99.51
0.170 0.796 340.0 101.9 0.3715 361.5 86.8 11.27
0.172 0.820 273.5 101.9 0.3040 296.5 83.2 9.44
0.175 0.853 187.3 101.9 0.2283 222.5 78.3 6.88
0.173 0.857 374.2 120.4 0.4055 387.2 111.2 9.67
0.172 0.885 302.5 120.4 0.3530 337.2 106.7 8.16
2.2334
2.3081
2.4475
2.4994
2.6192
0.0838
0.0811
0.0778
0.0779
0.0738 4.20
5.10
4.40
3.345
4.73
Cpb'
1.719
1.698
1.656
1.646
1.601
wep' k'L
12.05
9.54
6.32
12.43
9.87
1910
1788
1658
2318
1828
D L
ATar.Vpr-
6 88
INDUSTRIAL AND ENOINE6RINO CHEMISTRY
I t has been postulated that the conventional heat-transfer correlations could be used by employing effective thermal conductivities and specific heats. Equation 33 was used to check the measured heat-transfer coefficient of air in the turbulent range. Rearranged, the equation reads:
This equation is adequate for small temperature differences where variations in the physical properties are small. For large temperature differences, !Veiland and Lowdermilk (74) found that the film temperature should be used to calculate all the physical properties. In addition, a small effect of the lengthto-diameter ratio was observed. They
H E A T T R A N S F E R TO G A S - P H A S E R E A C T I O N obtained a successful correlation with the equation :
(g)
= 0.021
(T) DVbrj
x
0.8
where for our use the effective physical properties have been substituted. Equations 34 and 35 are similar, except for the temperature at which the physical properties are evaluated. Heat-transfer coefficients for the experimental nitrogen peroxide system in turbulent flow were calculated from Equation 35. Although large temperature differences were not encountered the reaction caused important variations in the physical properties, which should be accounted for. Values of specific heat and heat of dissociation were taken from the National Bureau of Standards (77). The viscosities of nitrogen dioxide and nitrogen tetroxide were estimated from an equation developed by Lane ( 9 ):
The normal boiling point of nitrogen peroxide, 295" K., was used for TB. Although this is actually the boiling point of the mixture, it was estimated from critical temperature, parachor, and vapor pressure data that the "boiling points" of nitrogen dioxide and nitrogen tetroxide would not be much different from the measured value of 22" C . for the mixture. The molecular volumes, uB, at the normal boiling point were calculated from the mass density: 31.9 cc./g. mole
U B ~= ~ ,
and
cc./g.~ mole ~
B = 63.8 ~ ~
U
Viscosities of the mixture could then be estimated from the equation of Bromley and Wilke (2, 75): P N O I - iiz04
+
PIT204
=
1
+-11.30a -a PNOa
0.75 (1
I +
or
- a ) centipoises
a
where the constants of 1.30 and 0.75 were read from curves given by Wilke and Bromley. It is estimated that the error in the calculation of the viscosity is less than 8%. This will cause an error of less than 1% in the predicted heat-transfer coefficient for the reacting system. The normal thermal conductivity can be calculated using the dimensionless relationship (70) :
K
In most instances, it is advisable to calculate local heat-transfer coefficients and to use a stepwise method for the determination of the desired tempera-, tures and conversions. Possible shortcut procedures depend on the experimental conditions. Calculated data are shown in Table I for the turbulent flow range. The measured and predicted heat-transfer coefficients are compared in Table 111. Good agreement has been obtained. The correlation of the data for the laminar flow range was somewhat complicated by the effect of natural convection, Usually, natural convection is significant only at very low gas flow rates. However, two effects contributed in the case of nitrogen peroxide: the high coefficient of volumetric expansion due to the volume increase by the dissociation reaction, and the high density of the gas mixture, leading to an increase in natural convection. Heat-transfer coefficients for laminar flow are usually expressed in the form:
where the effective physical properties have been substituted. The first group in the function on the right-hand side is the Graetz number, (wc,'/k'L), describing the flow rate; the second group, ( N Q , N ~ , D / L )expresses , the influence of natural convection. Some attempts have been made to write Equation 36 in the form of an analytical equation, but this has been successful only for large values of the Graetz number. The results have, therefore, been expressed in graphical form in Figure 3. All physical properties were evaluated at the customary bulk temperature of the gas, except pW. Again, Equation 13 was used to calculate the effective specific heat over the range from bulk inlet to bulk outlet temperatures. However, Equation 31 was used to determine
Table 111. Predicted and Measured Heat Trahsfer Coefficients for Turbulent Flow Nitrogen Peroxide Run No. 1 2 3 4
Heat-Transfer Coefficient B.T.U./ (Hi..)(54. Ft.) (" F.) Measured Predicted 68.5 45.5 58.8 56.9
70.2 44.7 57.0 56.5
the distance-average effective thermal conductivity over the same temperature range. Equation 28 for the temperature-average effective thermal conductivity gave nearly equal values. Calculated data are shown in Table 11. I n addition to the data points obtained in this research, the results of other investigators for heat transfer without a chemical reaction are also shown in Figure 3. Eubank and Proctor (5) compiled these data in an effort to arrive at a satisfactory correlation. While all the data for N G ~
