Heat area varies with volume

An exothermic reaction involving two reactants is run in a semi-continuous reactor. The heat evolution can be controlled by varying the feed rate of one component This is done via feedback control with reactor temperature measurement used to manipulate the feed rate. The reactor is cooled by a water jacket, for which the heat transfer area varies with volume. Additional control could involve the manipulation of the cooling-water flow rate. 

The heat transfer area varies with the square of the vessel diameter, and the volume varies with the cube of the vessel diameter. Thus the area-volume ratio (A/V) varies with volume as... 

Scale-up is non-geometric with length/diameter ratios varying from 2 1 to 30 1. The non-geometric scale-up helps to increase heat transfer area as reactor volume increases. 

Assuming Aq is the total heating surface in the full tank, with volume, Vq, and assuming a linear variation in heating area with respect to liquid depth, the heat transfer area may vary according to the simple relationship... 

If the rate of heat transfer to or from the broth is important, then the heat transfer area per unit volume of broth should be considered. As the surface area and the liquid volume will vary in proportion to the square and cube of the representative length of vessels, respectively, the heat transfer area of jacketed vessels may become insufficient with larger vessels. Thus, the use of internal coils, or perhaps an external heat exchanger, may become necessary with larger fermentors. 

The heat exchange area (A) may vary with time due to the volume increase by the feed. This variation is determined by the geometry of the reactor, especially by its height covered by the heat exchange system (jacket, internal coils, or welded half-coils). In case there is a significant change in the physical chemical properties of the reaction mixture, the overall heat exchange coefficient (U) will also be a function of time. 

In the normal case of a geometrically similar scale-up, it can be readily shown that the surface area per unit volume varies inversely with the vessel diameter. Thus larger vessels are more difficult to cool, since the heat generated by a reaction in a potential runaway situation is proportional to the vessel volume, whereas the surface area available to dissipate a given heat output is decreased. Vigorous reactions may require the reactor to be detuned by operating with more dilute feedstock in order to reduce the full-scale reaction intensity. This... 

The heat transfer area, 4hx varies with time because the volume of liquid in the vessel increases as feed is added. The instantaneous heat transfer area is calculated from the ratio of the instantaneous volume to the total volume ... 

In (his chapter, we considered the variation of temperature with time as well as position in one- or multidimensional sysKhls, We first considered the lumped systems in which tlte lempcr. itiire varies with time but remains uniform throughout the system at any time. The temperature of a lumped body of arbitrary shape of mass in, volume L/, surface area density p, and specific heat ioitially at a uniform temperature T, that is exposed to convection at time r 0 in a medium at temperature Tr. with a heal transfer coefficient h is expressed as... 

In practice, one is generally not so much interested in the linear growth rate Lc as in the amount (mass or volume) of crystals formed per unit volume and unit time. In principle, the latter can be given as Lc x Ac, where Ac is the specific surface area of the crystals. However, both factors tend to vary with time. Generally, Lc decreases because crystallization implies depletion of solute and hence a decrease of supersaturation. Moreover, release of the heat of fusion may cause the temperature to increase significantly, hence the solubility to increase, hence In [1 to decrease. Ac increases because (a) each crystal increases in size, and (b) more crystals are formed if nucleation goes on. Several, often complicated, growth rate theories have been worked out for various conditions. We will only touch on a few aspects. 

With practically all machines, only the cylinder temperature is directly controlled (see Chapter 1). The actual heat of the melt, within the screw and as it is ejected from the nozzle, can vary considerably, depending on the efficiency of the screw design and the method of operation. Factors affecting melt heat include the time plastic remains in the cylinder the internal surface heating area of the cylinder and the screw, per volume of material being heated the thermal conductivity of cylinder, screw, and plastic (Table 1-6) the heat differential between the cylinder and the melt and the amount of melt turbulence in the cylinder. In designing the screw, a balance must be maintained between the need to provide adequate time for heat exposure and the need to maximize output most economically. 

The extreme case of complete heat transfer control for COj-SA is illustrated in Figure 6.15. For this system diffusion is much faster and even in relatively large crystals the uptake rate is controlled by heat transfer. Uptake curves are essentially independent of crystal size but vary with sample size due to changes in the effective heat capacity and external area-to-volume ratio for the sample. Analysis of the uptake curves according to Eq. (6.70) yields consistent values for the overall heat capacity (34 mg sample C 0.32 and 12.5 mg sample 0.72 cal/g deg.). The variation of effective neat capacity with sample size arises from the increasing importance of the heat capacity of the containing pan when the adsorbent weight is small. 

In order to describe the dynamics of this system, we must include component material balances, an equation describing how the specific heat of the solution varies with the composition of the vessel, and the geometric relations which describe the heat-transfer area as a function of vessel volume. Since this is a binary mixture of components A and B, we can only write two material balances we choose to write one component balance and the overall mass balance. We will assume that the total mass density p is a constant for this system. Thus, the component mass balance is... 

Moderate intensity kLa of order 0.05 s for fast reactions with other slower steps where particle suspension and/or heat transfer require enhancement. Agitated vessels are useful here, and indeed are often selected where the intensity needs are uncertain, or may vary widely (as in general-purpose reactors). The larger top surface area per unit volume than can be achieved with bubble columns allows higher exit gas flow rates without liquid entrainment and carryover. 

It is assumed that the wall temperature varies with time only along its axial direction and is uniform throughout its thickness at any axial location. This condition is very closely obtained in any solid undergoing transient heat conduction when the Biot number, h (volume of solid/area wetted) /Jb, is less than 0.100. For the cases examined in this paper the maximum Biot number of the wall is 0.012. 

Then, the quantity of heat that could be removed in batch reactors whose volume varies from 11 to 1 m is calculated. In order to compare with experimental results, the temperature gradient is fixed at 45 °C (beyond which water in the utility stream would freeze and another cooling fluid should be used). The maximum global heat-transfer coefficient is estimated at an optimistic value of 500 W m K h The calculated value of the global heat transfer area of each batch reactor. A, is in the same range as the one given by the Schweich relation [35] ... 

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