Wall temperature, calculation

Consider what happens in the batch reactor given in Example 9.3.3 if the wall temperature does not remain constant. For comparison to the constant wall temperature, calculate the fractional conversion and reactor temperature as a function of time when ... 

This is so close to the wall temperature calculated previously that a second approximation is unnecessary. 

Ballal, D.R., and Lefebvre, A.H., A Proposed Method for Calculating Fibn Cooled Wall Temperatures in Gas Turbine Combustor Chambers, ASME Paper 72-WA/HT-24, 1972. 

The calculations are made as follows. The exchanger is divided into small increments to allow numerical integrations. A tube wall temperature is first calculated and then QAV. The gas temperature and composition from an increment can then be calculated. If the gas composition is above saturation for the temperature, any excess condensation can occur as a fog. This allows the degree of fogging tendency to be quantified. Whenever possible, experimental data should be used to determine the ratio of heat transfer to m.ass transfer coefficients. This can be done with a simple wet and dry bulb temperature measurement using the components involved. 

Heat transfer coefficient between a pipe and a wall. Water flows in a pipe d =15 mm) with a velocity of v = 1.0 m s. The mean temperature of water is 0 , = 15 °C, and the wall temperature 6 = 50 °C. Calculate the heat transfer coefficient away from the pipe inlet. For water the properties are... 

The air and wall temperatures and the concentrations in both zones are solved by iteration toward a steady-state situation or by simulating the time-dependent development. In the time-dependent calculation the heat capacity of the walls should be included. 

To calculate the outside film coefficient, you need to know the difference in temperature of the condensing vapor (T, ) and the pipe wall temperature (L). The pipe wall temperature is determined hy trial-and-error calculations using the following equation/ ... 

Calculate tube wall temperature, C. In this example ... 

When die gas is flowing at 5 m/s the thermocouple reads 323 K. When it is flowing at 10 ra/s it reads 313 K, and when it is flowing at 15.0 m/s it reads 309 K. Show that the gas temperature is about 298 K and calculate the approximate wall temperature. What temperature will the thermocouple indicate when the gas velocity is... 

Reynolds number. It should be stressed that the heat transfer coefficient depends on the character of the wall temperature and the bulk fluid temperature variation along the heated tube wall. It is well known that under certain conditions the use of mean wall and fluid temperatures to calculate the heat transfer coefficient may lead to peculiar behavior of the Nusselt number (see Eckert and Weise 1941 Petukhov 1967 Kays and Crawford 1993). The experimental results of Hetsroni et al. (2004) showed that the use of the heat transfer model based on the assumption of constant heat flux, and linear variation of the bulk temperature of the fluid at low Reynolds number, yield an apparent growth of the Nusselt number with an increase in the Reynolds number, as well as underestimation of this number. 

Optimization requires that a-rtjl have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that 3>AtlR be large. Thus, the method can be used for polymer systems that have thermal diffusivities typical of organic liquids but low molecular diffusivities. The calculations needed to solve the optimization are much longer than those needed to solve the ODEs of Chapter 6, but they are still feasible on small computers. 

To apply the correction an estimate of the wall temperature is needed. This can be made by first calculating the coefficient without the correction and using the following relationship to estimate the wall temperature ... 

Usually an approximate estimate of the wall temperature is sufficient, but trial-and-error calculations can be made to obtain a better estimate if the correction is large. 

If the degree of superheat is large, it will be necessary to divide the temperature profile into sections and determine the mean temperature difference and heat-transfer coefficient separately for each section. If the tube wall temperature is below the dew point of the vapour, liquid will condense directly from the vapour on to the tubes. In these circumstances it has been found that the heat-transfer coefficient in the superheating section is close to the value for condensation and can be taken as the same. So, where the amount of superheating is not too excessive, say less than 25 per cent of the latent heat load, and the outlet coolant temperature is well below the vapour dew point, the sensible heat load for desuperheating can be lumped with the latent heat load. The total heat-transfer area required can then be calculated using a mean temperature difference based on the saturation temperature (not the superheat temperature) and the estimated condensate film heat-transfer coefficient. 

Dispersed flow model. To calculate the actual quality, vapor temperature, and wall temperature, or heat flux, as functions of axial position beyond dryout... 

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat flux was specified as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ x 13-14. This is somewhat low for the k- model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. 

The temperature increase calculation in Sections 7.7.1 and 7.7.2 was based on the viscosity using the temperature at the entry to the metering section. Because the temperature of the resin increases as it flows downstream, the shear viscosity continuously decreases. A better method to calculate the temperature of the resin in the channel is to divide the channel into many Az or Az increments, and then for each increment, perform an energy balance on each control volume [67]. A schematic of the control volume is shown in Fig. 7.36. The energy balance includes convection into and out of the volume, dissipation due to rotation and pressure flows, and energy conduction through the barrel wall and the root of the screw. This section will describe a control volume method for temperature calculation for both screw rotation and barrel rotation. 

For cooling of non-Newtonian fluids, the pressure drop should be calculated by use of fluid properties at the wall temperature. 

An alternative method of calorimetry that gives less accurate results, but is simpler in concept, uses only a single insulated container and a thermometer. Temperature changes in the calorimeter are brought about by adding hot (or cold) objects of known weight and temperature. Calculations are based on the principle that the heat lost by the added hot object is equal to that gained by the water in the calorimeter and the calorimeter walls. This simple approach is illustrated in the next two problems. 

For the reformer we assume that the outer wall temperature profile of the reformer tubes decouples the heat-transfer equations of the furnace from those for the reformer tubes themselves. The profile is correct when the heat flux from the furnace to the reformer tube walls equals the heat flux from the tube walls to the reacting mixture. We must carry out sequential approximating iterations to find the outer wall temperature profile Tt,o that converges to the specific conditions by using the difference of fluxes to obtain a new temperature profile T) o for the outer wall and the sequence of calculations is then repeated. In other words, a T) o profile is assumed to be known and the flux Q from the furnace is computed from the equations (7.136) and (7.137), giving rise to a new Tt o-This profile is compared with the old temperature profile. We iterate until the temperature profiles become stationary, i.e., until convergence. 

EXAMPLE 15.2. The temperature of the inside wall of a tube is 200 °C and the outside wall temperature is 40 °C. Calculate the stresses at the outside of the wall if the tube is made from a glass having a coefficient of thermal expansion of a = 8 x 10 6/°C, an elastic modulus of 60 GPa, and a Poisson s ratio of 0.3. 

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