Heat transfer axial temperature diameter

From both of these equations, it will be noted that the heat transfer coefficient is noi a function of the temperature difference. Here Re" — (d N pj x-) and Re = (udvp/n), where dv is the tube diameter and u is the average velocity of the liquid in the film in the axial direction. 

However, the most complex analysis is that in which heat transfer through the reactor walls is taken into account. This type of operation must be employed when it is necessary to supply or remove energy from the system so as to moderate the temperature excursions that would otherwise follow. It is frequently employed in industrial reactors and, to model such systems, one must often resort to two-dimensional models of the reactor that allow the concentration and temperature to vary in both the radial and axial directions. In the analysis of such systems, we make incremental calculations across the diameter of a given longitudinal segment of the packed bed reactor, and then proceed to repeat the process for successive longitudinal increments. 

Heat transfer studies on fixed beds have almost invariably been made on tubes of large diameter by measuring radial temperature profiles (1). The correlations so obtained involve large extrapolations of tube diameter and are of questionable validity in the design of many industrial reactors, involving the use of narrow tubes. In such beds it is only possible to measure an axial temperature profile, usually that along the central axis (2), from which an overall heat transfer coefficient (U) can be determined. The overall heat transfer coefficient (U) can be then used in one--dimensional reactor models to obtain a preliminary impression of longitudinal product and temperature distributions. 

Stagnant saturated steam at 100°C condenses on the outside of a horizontal copper tube with outside diameter 5.0 cm. The copper tube has a wall thickness of 1.5 mm and a thermal conductivity of 390 W/m-K. At the axial location being considered, the bulk temperature of the vater flowing inside the copper tube is 80°C and the inside heat transfer coefficient is 3500 W/m2. Assuming film condensation, calculate the heat transfer and condensation rate per unit length of pipe. 

Theonly important current application of tubular reactors in polymer syntheses is in the production of high pressure, low density polyethylene. In tubular processes, the newer reactors typically have inside diameters about 2.5 cm and lengths of the order of I km. Ethylene, a free-radical initiator, and a chain transfer agent are injected at the tube inlet and sometimes downstream as well. The high heat of polymerization causes nonisothermal conditions with the temperature increasing towards the tube center and away from the inlet. A typical axial temperature profile peaks some distance down the tube where the bulk of the initiator has been consumed. The reactors are operated at 200-300°C and 2000-3000 atm pressure. 

Advantages of three-phase fluidized beds over trickle beds and other fixed bed systems are temperature uniformity, high heat transfer, ability to add and remove catalyst particles continuously, and limited mass transfer resistances (both external to the particles and bubbles, because of turbulence and limited bubble size, and inside the particles owing to relatively small particle diameters). Disadvantages include substantial axial dispersion (of gas, liquid, and particles), causing substantial deviations from plug flow, and lack of predictability because of the complex hydrodynamics. There are two major applications of gas-liquid-solid-fluidized beds biochemical processes and hydrocarbon processing. 

The objective is to solve Eqs. (13-42) and (13-20) for the temperature and conversion at any point in the catalyst bed. As boundary conditions at the entrance we need the temperature and conversion profile across the diameter of the reactor. Further boundary conditions applicable at any axial location are that the conversion is flat (dx/dr = 0) at both the centerline and the wall of the tube. The temperature gradient at the centerline is zero, but the condition at the wall is determined by the heat-transfer character-... 

A vertical boiler pipe of inner radius Rt and outer radius Rq is subjected to a uniform heat flux q" (Fig. 2P-4). A water flow evaporates through this pipe. The inside heat transfer coefficient h is very large. Neglecting axial conduction in the pipe, (a) evaluate the local -temperature difference between the inner and onter surfaces of the pipe, (b) evaluate the local difference between the bulk temperature of the water and the inner surface temperature of the pipe. Data Pipe diameters Do = 12 cm and Di = 10 cm. Furnace temperature Tg = 1400 ° C Evaporating water pressure in the pipe p = 10 MPa. Thermal conductivity of steel pipe k = 20 W/m-K. 

Due to the forced aeration, oxygen supply in packed bed bioreactors is usually not a problem, even if quite low aeration rates are used [138]. In contrast, temperature control can be difficult, especially in packed beds above 15 cm diameter and lacking internal heat transfer plates. In such bioreactors, the main heat transfer mechanisms are axial convection and evaporation, and radial temperature gradients are negUgible except close to the bioreactor wall. The dynamics of convective cooHng mean that axial temperature gradients are established and temperatures over 20 °C higher than the inlet air temperature... 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z < 0), the fluid temperature is uniform at Tf for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

---