Constant Wall Temperature

At constant wall temperature, the asymptotes from theory are predicted as follows  

With the Hausen correlation [13], Nusselt numbers in the thermal entry length can be calculated  

For a combined entry length, the Sieder-Tate [62] correlation is suitable  

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Circular Tubes For horizontal tubes and constant wall temperature, several relationships are available, depending on the Graetz number. For 0.1 < Ngz < 10 Hausens [A//g. Waermetech., 9, 75 (1959)], the following equation is recommended. 

Constant wall temperature Constant heat flux... 

Parallel Plates and Rectangular Ducts The limidng Nusselt number for parallel plates and flat rectangular ducts is given in Table 5-4. Norris and Streid [Tran.s. Am. Soe. Meeh. Eng., 62, 525 (1940)] report for constant wall temperature... 

The thermos phon circulation rate can be as high as 10 to 15 times the coolant evaporation rate. This, in turn, eliminates any significant temperature difference, and the jacket is maintained under isothermal conditions. In this case, the constant wall temperature assumption is satisfied. During starting of the thermosiphon, the bottom can be 20-30°C hotter, and the start of circulation can be established by observing that the difference between the top and bottom jacket temperature is diminishing. Figure 2.2.5 (Berty 1983) shows the vapor pressure-temperature relationship for three coolants water, tetralin, and Dowtherm A. 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

Uniform liquid temperature distribution at the inlet, constant wall temperature... 

Unlike at adiabatic conditions, the height of the liquid level in a heated capillary tube depends not only on cr, r, pl and 6, but also on the viscosities and thermal conductivities of the two phases, the wall heat flux and the heat loss at the inlet. The latter affects the rate of liquid evaporation and hydraulic resistance of the capillary tube. The process becomes much more complicated when the flow undergoes small perturbations triggering unsteady flow of both phases. The rising velocity, pressure and temperature fluctuations are the cause for oscillations of the position of the meniscus, its shape and, accordingly, the fluctuations of the capillary pressure. Under constant wall temperature, the velocity and temperature fluctuations promote oscillations of the wall heat flux. 

Example 8.9 Find the temperature distribution in a laminar flow, tubular heat exchanger having a uniform inlet temperature and constant wall temperature Twall- Ignore the temperature dependence of viscosity so that the velocity profile is parabolic everywhere in the reactor. Use art/P = 0.4 and report your results in terms of the dimensionless temperature... 

Hsu and Graham (1961) took into consideration the bubble shape and incorporated the thermal boundary-layer thickness, 8, into their equation, thus making the bubble growth rate a function of 8. Han and Griffith (1965b) took an approach similar to that of Hsu and Graham with more elaboration, and dealt with the constant-wall-temperature case. Their equation is... 

In this section a short description of a comparison between experimental and simulation results for heat transfer is illustrated (Nijemeisland and Dixon, 2001). The experimental set-up used was a single packed tube with a heated wall as shown in Fig. 8. The packed bed consisted of 44 one-inch diameter spheres. The column (single tube) in which they were packed had an inner diameter of two inches. The column consisted of two main parts. The bottom part was an unheated 6-inch packed nylon tube as a calming section, and the top part of the column was an 18-inch steam-heated section maintained at a constant wall temperature. The 44-sphere packed bed fills the entire calming section and part of the heated section leaving room above the packing for the thermocouple cross (Fig. 8) for measuring gas temperatures above the bed. 

When we want to look at the connection between the flow behavior and the amount of heat that is transferred into the fixed bed, the 3D temperature field is not the ideal tool. We can look at a contour map of the heat flux through the wall of the reactor tube. Fig. 19 actually displays a contour map of the global wall heat transfer coefficient, h0, which is defined by qw — h0(Tw-T0) where T0 is a global reference temperature. So, for constant wall temperature, qw and h0 are proportional, and their contour maps are similar. The map in Fig. 19 shows the local heat transfer coefficient at the tube wall and displays a level of detail that would be hard to obtain from experiment. The features found in the map are the result of the flow features in the bed and the packing structure of the particles. 

A liquid-phase reaction A + B -> 2C is conducted in a nonisothermal multitubular PFR. The reactor tubes (7 m long, 2 cm in diameter) are surrounded by a coolant which maintains a constant wall temperature. The reaction is pseudo-first-order with respect to A, with kA = 4.03 X 105 e 56WT s 1. The mass flow rate is constant at 0.06 kg s-1, the density is constant at 1.025 g cm-3, and the temperature at the inlet of the reactor (TJ is 350 K. 

The following model is proposed for the oxidation of naphthalene in a tubular flow reactor with constant wall temperature (Welsenaere Froment, Chem Eng Science 25 1503, 1970) ... 

Shown in Fig. 7.2 is the relationship between qr and qL for various initial pressures, a value of the heat transfer coefficient h, and a constant wall temperature of In Eq. (7.8) qr takes the usual exponential shape due to the Arrhenius kinetic rate term and cp is obviously a linear function of the mixture temperature T. The qt line intersects the qr curve for an initial pressure l at two points, a and b. 

It has been demonstrated that kg can be estimated by analogy with the Graetz-Nusselt problem governing heat transfer to a fiuid in a duct with constant wall temperature (SH= Nut) [30] and that the axial concentration profiles of NO and of N H 3 provided by the 1D model are equivalent and almost superimposed with those of a rigorous multidimensional model of the SCR monolith reactor in the case of square channels and of ER kinetics, which must be introduced to comply with industrial conditions for steady-state applications characterized by substoichiometric NH3 NO feed ratio, that is, a[Pg.401]

Constant Wall Temperature Case (NuJRa 4 versus Pr)... 

For the constant wall temperature case this reduces to ... 

In real systems, the increase of temperature is accompanied by a corresponding increase of pressure, which may lead to an explosion (i.e., to an uncontrolled increase of pressure). Nevertheless, the analysis of temperature patterns with simple kinetics is enough to study the problem for adiabatic reactors and for constant wall temperature (isoperibolic) reactors, whereas the more complex case of controlled wall temperature requires the adoption of more advanced methods. 

The assumption of constant wall temperature is often more realistic for chemical reactors than the adiabatic case. In this respect, starting from the pioneering theory of thermal explosions developed by Semenov at the beginning of the last century [8], significant advances have been made by the related scientific literature with approaches that can be roughly classified as geometric and sensitivity-based, as described in detail in the following. 

Table 9.3 Average Nu for uniform wall heat flux and constant wall temperature, Ostrach [3].

If the solution procedure is carried through as outlined above, the following is obtained for fully developed laminar flow through a pipe with constant wall temperature ... 

Since the case of a constant wall temperature is being considered, i.e., since 0i,N = 0 and since the symmetry condition on the center line gives, in finite-difference form, 0,i = 0, 2, the following two results are obtained by considering the form of Eq. (4.i55) ... 

Azer, N.Z.,and Chao, B.T., Turbulent Heat Transfer in Liquid Metals—Fully Developed Pipe Row with Constant Wall Temperature , Int. J. Heat Mass Transfer, Vol. 3, p. 77, 1961. 

Jove. D.D., Correlation for Opposing Flow, Mixed Convection Heat Transfer in a Vertical Tube with Constant Wall Temperature . J. Heat Transfer. Vol. 118, pp. 787-789, 1996. 

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