Temperature driving force

At,n = log-mean-temperature driving force from heating medium to the sohds, K. 

The temperature driving force for drying is the difference between the drying-gas outlet temperature and, in the case of pure water, the gas wet-bulb temperature. In the case of a solution, the adiabatic saturation temperature of the pure saturated solution is employed rather than the wet-bulb temperature. 

Kj= thermal conductivity of gas film surrounding the droplet, Btu/(h ft )(°F ft), evaluated at the average between diyer gas and drop temperature V = volume of diyer chamber, rP At = temperature driving force (under terminal conditions described above), °F D, = maximum drop diameter, ft to, = weight rate of liquid flow, Ib/h p, = density of hquid, Ib/ft ... 

Equations (13-115) to (13-117) contain terms, for rates of heat transfer from the vapor phase to the hquid phase. These rates are estimated from convective and bulk-flow contributions, where the former are based on interfacial area, average-temperature driving forces, and convective heat-transfer coefficients, which are determined from the Chilton-Colburn analogy for the vapor phase and from the penetration theoiy for the liquid phase. 

For counter-current flow of the fluids through the unit with sensible heat transfer only, this is the most efficient temperature driving force with the largest temperature cross in the unit. The temperature of the outlet of the hot stream can be cooler than the outlet temperature of the cold stream, see Figure 10-29 ... 

Johnson, A. 1., Circulation Rates and Over-All Temperature Driving Forces in a Vertical Thermosiphon Reboiler, presented at AlChE Heat Transfer Sym., Louisville, KY, Mar. 20 (1955). 

It may be noted that using Underwood s approximation (equation 9.10), the calculated values for the mean temperature driving forces are 41.9 K and 39.3 K for counter- and co-current flow respectively, which agree exactly with the logarithmic mean values. 

Equation 11.72 cannot be integrated directly, however, because the temperature driving force (0j — 0O) is not known as a function of location x on the plate. The solution of equation 11.72 involves a quite complex procedure which is given by KAYS and CRAWFORD 45 and takes the following form ... 

Thus, the minimum value for UAext pQCp is about 5. If the heat transfer group is any smaller than this, stable operation at 7), , = 413K by manipulation of Text is no longer possible because the temperature driving force, AT = Tout — Text, becomes impossibly large. As will be seen in Section 5.3.2, the quantity UAext pQCp declines on a normal scaleup. 

These results are summarized in the last four rows of Table 4.1. Scaling the volume by a factor of 512 causes a large loss in hAg t per unit volume. An increase in the temperature driving force (e.g., by reducing by a factor of 10 could compensate, but such a large increase is unlikely to be possible. Also, with cooling at the walls, the viscosity correction term in Equation (5.34) will become important and will decrease hAg t still more. 

Example 5.11 The results of Table 5.1 suggest that scaling a tubular reactor with constant heat transfer per unit volume is possible, even with the further restriction that the temperature driving force be the same in the large and small units. Find the various scaling factors for this form of scaleup for turbulent liquids and apply them to the pilot reactor in Example 5.10. 

Applying these factors to the 5= 128 scaleup in Example 5.10 gives a tube that is nominally 125 = 101 ft long and 1.0495 = 4.1 inches in diameter. The length-to-diameter ratio increases to 298. The Reynolds number increases to 85005 = 278,000. The pressure drop would increase by a factor of 0.86 j jjg temperature driving force would remain constant at 7°C so that the jacket temperature would remain 55°C. 

Solution Now, Ar=107°C. Scaling with geometric similarity would force the temperature driving force to increase by S = 1.9, as before, but the scaled-up value is now 201°C. The coolant temperature would drop to —39°C, which is technically feasible but undesirable. Scaling with constant pressure forces an even lower coolant temperature. A scaleup with constant heat transfer becomes attractive. 

This section has based scaleups on pressure drops and temperature driving forces. Any consideration of mixing, and particularly the closeness of approach to piston flow, has been ignored. Scaleup factors for the extent of mixing in a tubular reactor are discussed in Chapters 8 and 9. If the flow is turbulent and if the Reynolds number increases upon scaleup (as is normal), and if the length-to-diameter ratio does not decrease upon scaleup, then the reactor will approach piston flow more closely upon scaleup. Substantiation for this statement can be found by applying the axial dispersion model discussed in Section 9.3. All the scaleups discussed in Examples 5.10-5.13 should be reasonable from a mixing viewpoint since the scaled-up reactors will approach piston flow more closely. 

The important quantities in this term are the heat transfer area A, the temperature driving force or difference (Tg-Ti), where Ta is the temperature of the heating or cooling source, and the overall heat transfer coefficient U. The heat transfer coefficient, U, has units of (energy)/(time)(area)(degree), e.g., J/s m K. The units for U A AT are thus... 

The cost of recovery will be reduced if the streams are located conveniently close. The amount of energy that can be recovered will depend on the temperature, flow, heat capacity, and temperature change possible, in each stream. A reasonable temperature driving force must be maintained to keep the exchanger area to a practical size. The most efficient exchanger will be the one in which the shell and tube flows are truly countercurrent. Multiple tube pass exchangers are usually used for practical reasons. With multiple tube passes the flow will be part counter-current and part co-current and temperature crosses can occur, which will reduce the efficiency of heat recovery (see Chapter 12). 

Waste-heat boilers are often used to recover heat from furnace flue gases and the process gas streams from high-temperature reactors. The pressure, and superheat temperature, of the stream generated will depend on the temperature of the hot stream and the approach temperature permissible at the boiler exit (see Chapter 12). As with any heat-transfer equipment, the area required will increase as the mean temperature driving force (log mean AT) is reduced. The permissible exit temperature may also be limited by process considerations. If the gas stream contains water vapour and soluble corrosive gases, such as HC1 or S02, the exit gases temperature must be kept above the dew point. 

A Tm = the mean temperature difference, the temperature driving force, °C. 

Figure 19.6 Nonisothermal mixing degrades temperature driving forces and might transfer heat across the pinch.

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