Time constant of heat transfer

The time constant of the heat transfer through the glass wall is calculated from the physical properties of the wall [25], 

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In the above equations, Cpr and Cp< denote heat capacities of the fluid and solid phases, pb is the bed density and hp is the heat transfer coefficient between fluid and particles. Transport of heat through the fluid phase in the axial direction and in the radial direction of the bed by conduction are described by the effective thermal conductivities, ka,i and kas, while in the solid phase thermal conduction can be assumed to be isotropic and the effective thermal conductivity ka can be used to express this effect. Q i represents the heat evolution/absorption by adsorption or desorption on the basis of bed volume. This model neglects the temperature distribution in the radial position of each particle, which may seem contradictory to the case of mass transfer, where intraparticle mass transfer plays a significant role in the overall adsorption rate. Usually in the case of adsorption, the time constant of heat transfer in the particle is smaller than the time constant of intraparticle diffusion, and the temperature in the particle may be assumed to be constant. 

In order to achieve bulk gas separation by adsorption, the adsorbent must be used repeatedly. The desorption step takes a rather long time if thermal desorption is employed because of a relatively large time constant of heat transfer due to poor thermal conduction in the adsorbent packed bed. 

The time constants characterizing heat transfer in convection or radiation dominated rotary kilns are readily developed using less general heat-transfer models than that presented herein. These time constants define simple scaling laws which can be used to estimate the effects of fill fraction, kiln diameter, moisture, and rotation rate on the temperatures of the soHds. Criteria can also be estabHshed for estimating the relative importance of radiation and convection. In the following analysis, the kiln wall temperature, and the kiln gas temperature, T, are considered constant. Separate analyses are conducted for dry and wet conditions. 

It is frequently required to examine the combined performance of two or more processes in series, e.g. two systems or capacities, each described by a transfer function in the form of equations 7.19 or 7.26. Such multicapacity processes do not necessarily have to consist of more than one physical unit. Examples of the latter are a protected thermocouple junction where the time constant for heat transfer across the sheath material surrounding the junction is significant, or a distillation column in which each tray can be assumed to act as a separate capacity with respect to liquid flow and thermal energy. 

Additionally, we assume that the time constants for heat transfer and mass transport are of the same order of magnitude, i.e.,... 

The reactor is rather well behaved when the outside waU temperature is 335 K, but thermal runaway occurs when Twau = 336 K. Hence, Twaii exhibits a critical value between 335 and 336 K because thermal runaway occurs when Twaii > (rwaii)criticai- Thermal runaway can be prevented when Twau = 340 K if the surface-to-volume ratio of the reactor is increased by decreasing the tube radius R. This important design modification is accounted for by decreasing the time constant for heat transfer across the lateral surface. Numerical results are summarized in Table 4-2 for a single-pipe reactor with Tuaet = Rwaii = 340 K at nine different values of the heat transfer time constant. 

The time constant for heat transfer across the inner wall of the double-pipe configuration is the same for the reactive fluid and the nonreactive cooling fluid ... 

The characteristic time constants of heat and mass transfer strongly increase with decreasing tube diameter (Table 4.10.10), whereas the specific volumetric surface area ay increases with decreasing tube radius, as Oy is inversely proportional to rc-... 

A guarded hot-plate method, ASTM D1518, is used to measure the rate of heat transfer over time from a warm metal plate. The fabric is placed on the constant temperature plate and covered by a second metal plate. After the temperature of the second plate has been allowed to equiUbrate, the thermal transmittance is calculated based on the temperature difference between the two plates and the energy required to maintain the temperature of the bottom plate. The units for thermal transmittance are W/m -K. Thermal resistance is the reciprocal of thermal conductivity (or transmittance). Thermal resistance is often reported as a do value, defined as the insulation required to keep a resting person comfortable at 21°C with air movement of 0.1 m/s. Thermal resistance in m -K/W can be converted to do by multiplying by 0.1548 (121). 

The thermal design of tank coils involves the determination of the area of heat-transfer surface required to maintain the contents of the tank at a constant temperature or to raise or lower the temperature of the contents by a specified magnitude over a fixed time. 

In the above example, 1 lb of initial steam should evaporate approximately 1 lb of water in each of the effects A, B and C. In practice however, the evaporation per pound of initial steam, even for a fixed number of effects operated in series, varies widely with conditions, and is best predicted by means of a heat balance.This brings us to the term heat economy. The heat economy of such a system must not be confused with the evaporative capacity of one of the effects. If operated with steam at 220 "F in the heating space and 26 in. vacuum in its vapor space, effect A will evaporate as much water (nearly) as all three effects costing nearly three times its much but it will require approximately three times as much steam and cooling water. The capacity of one or more effects in series is directly proportional to the difference between the condensing temperature of the steam supplied, and the temperature of the boiling solution in the last effect, but also to the overall coefficient of heat transfer from steam to solution. If these factors remain constant, the capacity of one effect is the same as a combination of three effects. 

In the case of a temperature probe, the capacity is a heat capacity C == me, where m is the mass and c the material heat capacity, and the resistance is a thermal resistance R = l/(hA), where h is the heat transfer coefficient and A is the sensor surface area. Thus the time constant of a temperature probe is T = mc/ hA). Note that the time constant depends not only on the probe, but also on the environment in which the probe is located. According to the same principle, the time constant, for example, of the flow cell of a gas analyzer is r = Vwhere V is the volume of the cell and the sample flow rate. 

In the problems which have been considered so far, it has been assumed that the conditions at any point in the system remain constant with respect to time. The case of heat transfer by conduction in a medium in which the temperature is changing with time is now considered. This problem is of importance in the calculation of the temperature distribution in a body which is being heated or cooled. If, in an element of dimensions dr by dy by dr (Figure 9.9), the temperature at the point (x, y, z) is 9 and at the point (x + dx, y + dy, r. + dr) is (9 4- d6>), then assuming that the thermal conductivity k is constant and that no heat is generated in the medium, the rate of conduction of heat through the element is ... 

In the case of a storage tank with liquor of mass m and specific heat C heated by steam condensing in a helical coil, it may be assumed that the overall transfer coefficient U is constant. If 7 is the temperature of the condensing steam, Tt and To the initial and final temperatures of the liquor, and A the area of heat transfer surface, and T k the temperature of the liquor at any time t, then the rate of transfer of heat is given by ... 

We follow the analysis of Frank-Kamenetskii [3] of a slab of half-thickness, rG, heated by convection with a constant convective heat transfer coefficient, h, from an ambient of Too. The initial temperature is 7j < 7 ,XJ however, we consider no solution over time. We only examine the steady state solution, and look for conditions where it is not valid. If we return to the analysis for autoignition, under a uniform temperature state (see the Semenov model in Section 4.3) we saw that a critical state exists that was just on the fringe of valid steady solutions. Physically, this means that as the self-heating proceeds, there is a state of relatively low temperature where a steady condition is sustained. This is like the warm bag of mulch where the interior is a slightly higher temperature than the ambient. The exothermiscity is exactly balanced by the heat conducted away from the interior. However, under some critical condition of size (rG) or ambient heating (h and Too), we might leave the content world of steady state and a dynamic condition will... 

In this section, the analysis of the data reconciliation problem is restricted to quasi-steady-state process operations. That is, those processes where the dominant time constant of the dynamic response of the system is much smaller than the period with which disturbances enter the system. Under this assumption the system displays quasi-steady-state behavior. The disturbances that cause the change in the operating conditions may be due to a slow variation in the heat transfer coefficients, catalytic... 

Biochemical processes such as protein unfolding/refold-ing and supramolecular assembly/disassembly take place on a time scale of seconds to minutes after readjusting the temperature of a system. Most commercially available glass-jacketed cuvettes are not suitable for temperature jumps on this time scale, as a result of the slow kinetics of heat transfer across substances with characteristically high dielectric constants, and their use can convolute the time scale of the temperature change onto the time scale... 

In the adsorption microcalorimetry technique, the sample is kept at a constant temperature, while a probe molecule adsorbs onto its surface, and a heat-flow detector emits a signal proportional to the amount of heat transferred per unit time. 

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