20 Basic Models of Simultaneous Heat and Mass Transfer Keywords: Unit Models, Evaporator, Vaporizer A chemical process invariably involves energy transfer simultaneously with mass transfer. So in this part basic concepts of energy balance for simple flow systems are developed. CASE: 1 Figure shows a vessel with the steam jacket, an inlet flow, /time., and an outlet flow, volume The holdup in the vessel, V varies according to the equation The energy balance for the contents of the vessel is similar to the mass balance, that is Rate of change of heat energy in vessel heat in heat out Heat content of vessel VcϕT2; heat flowing into vessel heat flowing out of vessel heat transferred from jacket q UA Where cϕt2, cϕt2 temperature of inlet flow Temperature of contents of vessel, U overall heat transfer coefficient across jacket wall; A jacket wall area, Steam temperature in jacket wall; c specific heat of fluid, ϕ density
Fig. 20.1: Open Vessel with Stream Jacket By substituting these terms in the energy balance statement we obtain the following equation: The temperature of the steam jacket is a function of the pressure, and it is assumed that this pressure is controlled to a known value. The steam temperature can be defined merely as a function of pressure; that is, Heat balance Jacket flux
Fig. 20.2: Model for Stream Jacketed Vessel CASE: 2 In this example two complications are added the first I instead of single feed there are two feed flows, FA and FB, each with the different specific heat ca and cb. The second complication is to assume that the heat transfer area A between the steam jacket and the content of the vessel varies significantly because of the variation in level. Density variations are negligible. The specific heat of the vessel is Where C concentration Φ density Two more equation is required to establish the concentration CA and CB. Rate of accumulation inflow outflow The variation in volume V will cause the heat transfer area A to vary according to the following equation: Where D diameter of vessel. 4 4
Fig. 20.3: Stream Jacket Open Vessel Two inlet Stream with Mixing
Component mass balance for A Jacket heat flux Area 4 4 Specific heat Heat balance Component mass balance for B Fig. 20.4: Model of Fig. 20.3 BOILING Supposed a container of fluid is heated at a rate q Vol./time. A heat balance equation would state rate of change of heat content heat in heat out 0 no heat loss Where V volume c specific heat
q0 Fig. 20.5 Heat Balance Model Fig. 20.6: Vapor Pressure / temperature relationship Heat balance q Vapor pressure Temperature Equilibrium v for >0 Fig. 20.7: Model Microscopic for Equilibrium Balance. Total pressure π Heat Flux q T temperature Boiling System Fig. 20.8: Macroscopic Input- Output Relationship V Vapor flux
q Fig. 20.9: Macroscopic Model for Boiling Case: 3 Boiling in enclosed Vessel Suppose a vessel containing a single component fluid is totally enclosed with a gas space above the fluid. If this system is heated the temperature rises until boiling starts, after which the vapor evolved accumulates in the gas space and consequently raises the pressure. In turn, this increases in pressure forces the boiling temperature to rise. Such a system is sometimes erroneously viewed as the rise in temperature causing rise in pressure. Clearly, the pressure rises because more vapor is forced into the gas volume, a process that can be defined by using the gas law for ideal gases, PVG mrt q Fig. 20.10: Use of Heat Balance for boiling System
In this case it is necessary to include the change in sensible heat of the liquid mass. The heat balance equation on the fluid mass becomes Where q is the vapor enthalpy. Differentiating by parts and substituting equation reduces to q Fluid mass q Heat balance Vapor mass Gas space PVG mrt Fig. 20.11: Model of a Boiling Heat Transfer enclosed vessel, the
Fig. 20.12: Continuous Flow Boiling System Gas pressure Valve 1 P Vapor mass PVG mrt Heat Balance q Fig. 20.13: Model for Continuous Flow, Boiling Jacketed Vessel