# 快速时时彩: Low temperature fuel cells.pdf

Low temperature fuel cells J. Divisek Institute for Materials and Processes in Energy Systems, Forschungszentrum J¨ulich GmbH, J¨ulich, Germany 1 ENERGY BALANCE AND HEAT TRANSFER IN FUEL CELLS The principle of energy conservation determines the energy balance equation, which can be generally formulated as the sum of single rates of energy input ? energy ouput + energy production = accumulation of energy (1) This balance equation is true for a particular system or control volume. In a fuel cell, the balance takes into account the thermal and the electrical energy. Enthalpy change, Delta1H, within a closed constant molar volume system, V , can be written as Delta1H = Q mol ? W + VDelta1P (2) In equation (2), Q mol denotes the molar heat supplied to the system, W is the work done and P is the system pressure. An electrochemical system, however, is an open one. For this kind of system, the thermal energy balance is based on the difference between the enthalpy flow into the cell and the flow out of the cell. An accumulation of the thermal energy generally occurs during the changes from one operating condition to another, e.g., during start- up or shut-down. The time variation of energy balances is usually manifested in temperature changes. The energy input into the cell is associated with the enthalpy content of the inflowing process fluids, the fuel and the oxidant. The major contribution of the input energy is associated with the reaction enthalpy of the fuel combusting reaction. The rate of enthalpy outflow is given by the enthalpy content of the outgoing process fluids, the joule heat and heat losses from the reactor surfaces. An important issue of a fuel cell reaction is the fact that, in contrast to the usual chemical combustion, the complete fuel cell reaction occurs at two different electrodes, i.e., we have to deal with the delocalization of the chemical energy source. This may become a complication, especially when concerning the estimation of the local entropy sources, which are usually known only for the reaction as a whole. When we consider a fuel cell as a whole, it is possible to formulate the following general energy balance equation for the accumulation of energy in the cell (left-hand side of equation (3)) in the form summationdisplay m i c pi dT dt =?I parenleftbigg Delta1H nF + E c parenrightbigg + summationdisplay N i Ac pi (T in ? T out ) ? Q (?) E c = E 0 ? summationdisplay |η|?IR i ,E 0 =? Delta1G nF (3) In equation (3), A and I are the total system area and the total current flow, respectively, Delta1H is the enthalpy change due to the fuel cell reaction, summationtext N i Ac pi (T in ? T out ) is the enthalpy change between the inlet and outlet for all the components involved (T out being generally the cell temperature, T ), N i is the mass flow of each component, m i the mass of each component, E c the cell voltage and Q (?) are the enthalpy heat losses in the cell. E 0 is the voltage without current passing, usually termed as the electro- motive force (also the open circuit voltage). Handbook of Fuel Cells – Fundamentals, Technology and Applications ? 2010 John Wiley coordinates. (a) Cartesian, (b) cylindrical and (c) spherical. Table 1. Thermal conductivity of various materials at 0 ? C. Material Thermal conductivity (λ,Wm ?1 K ?1 ) Iron (pure) 73 Carbon steel (1% C) 43 Chrome-nickel steel (18% Cr, 8% Ni) 16 Copper (pure) 385 Graphite 170 Ceramic 2 Glass 0.80 Water 0.56 Hydrogen gas 0.175 Air 0.024 Water vapor (saturated) 0.021 Carbon dioxide 0.015 Thermal layer Flow Free streem Velocity layer δ(x) δ t (x ) U(∞) T s T f T T ∞ Figure 3. Velocity and thermal boundary layers. some of the principles of fluid dynamics necessary. The most fundamental principle in this respect is the boundary layer concept. The thermal boundary layer can be defined similarly to the velocity boundary layer. It concerns the region where the effect of the wall on the motion of the fluid 4 Low temperature fuel cells is significant (cf. Figure 3). Outside the boundary layer, it is assumed that the effect of the wall may be neglected. The limit of the layer is usually taken to be at the distance from the wall at which the fluid velocity is equal to the main stream value, being approximately between 95 and 99%. The flow conditions can be both laminar or turbulent. If the solid is maintained at a temperature T s ,whichis different from the fluid temperature T f ,avariationinthe fluid temperature is observed, i.e., the fluid temperature changes from T s to T f far away from the wall, designated as T f∞ , the temperature of the free stream as illustrated in Figure 3. The thermal boundary layer is generally not coincident with the velocity boundary layer. Even if the physical mechanism of heat transfer at the wall itself is a conduction process, the temperature gradient on the wall is dependent on the rate at which the fluid carries the heat away and is a function of the flow field. The whole process is therefore very complex. All the complexities involved in equation (11) may be lumped together in terms of single parameters by the introduction of Newton’s law of cooling in the simple form Q = hA(T s ? T f∞ )(12) The quantity h is the convection heat transfer coefficient or unit thermal conductance. This means that this coefficient is not a material property like the thermal conductivity λ. It is a complex function of the composition of the fluid, the geometry of the solid surface and the hydrodynamic conditions in the surroundings of the surface. Usually, it must be determined experimentally. The typical range of convection heat transfer coefficient values is indicated in Table 2. If we compare equations (5) and (12), we recognize that the temperature gradient ?T/?x is at least partially included in the heat transfer coefficient, h.FromTable2, it can be anticipated that, in contrast to the thermal conductivity, the heat transfer coefficient can only be Table 2. Approximate values of convection heat transfer coefficient. Mode Heat transfer coefficient (h,Wm ?2 K ?1 ) Free convection in air 5–25 Free convection in water 500–1000 Forced convection in air 10–500 Forced convection in water 100–15 000 Boiling water in a container 2500–35 000 Boiling water flowing in a tube 5000–100 000 Condensation of water vapor 5000–25 000 estimated approximately within some range, which in fact reveals the whole complexity included in this quantity. 1.2 Heat transfer without change of state Newton’s law of convective heat transfer, e.g., equa- tion (12), postulates a linear relationship between the heat flux and the temperature difference between the wall and the bulk of the fluid. It may be assumed that both phases have the same temperature at their boundary, which varies from the value at the wall to that in the bulk across the thermal boundary layer. In the immediate vicinity of the boundary, heat is transferred through the fluid flow perpen- dicularly to the wall by conduction. For laminar flow, only the molecular conductivity contributes to this process. In turbulent flows, a very thin viscous laminar sublayer exists in which the molecular conduction of heat occurs. Beyond this layer, velocity fluctuations normal to the main stream direction predominantly affect the heat exchange, which leads to an increase of the effective conductivity by orders of magnitude. Thus, the main heat resistance in the case of turbulent flow exists in the viscous sublayer. Therefore, the temperature gradient is very high near the wall and flat in the bulk. For laminar flow the temperature profile is well rounded and the heat conduction is also significant inside the fluid, but if the main flow is turbulent, a prac- tically constant temperature T f∞ can be calculated within the fluid. In both cases the driving temperature differences, Delta1T , in equation (12) results from the difference between the wall temperature and the bulk temperature averaged across the cross section. In turbulent flow the bulk tempera- ture is close to the temperature determined at some distance from the wall. For this case, Figure 4 schematically shows the temperature profile. The form of the temperature profile in Figure 4 can, in principle, be derived from the relation λ parenleftbigg ?T ?x parenrightbigg wall ≈ λ fl parenleftbigg ?T ?x parenrightbigg inner (13) The effect of the high value of the coefficient λ fl in the inner of a turbulent flow is that in the steady-state case no high temperature gradients can exist there, so that (?T/?x) wall will be higher than (?T/?x) inner by some decades. Equation (12) can also be used for laminar core flow in which there is no uniform internal temperature T f∞ .T f∞ can then be defined, for example, as the fluid temperature aver- aged over the entire flow cross-section, but other definitions are also possible. If formulas for the convection heat trans- fer coefficient h are used, the respective definition of T f∞ must be taken into consideration because it influences the 5 Heat transfer in fuel cells T emper ature T Phase 1 [fluid] Phase 2 Distance X T I T G Q ? Phase boundary ?T ?X Z Figure 4. Temperature profiles for heat transfer at turbulent flow conditions; T I : inner fluid temperature, T G : boundary wall temperature. value of h. In the contrary, an estimation can be made for h for a turbulent core flow, i.e., at practically constant T f∞ on the basis of a simple model assuming pure heat conduction in a wall zone of the fluid, which – provided that the wall is not curved too much – results in a nearly linear temperature profile in the laminar sublayer which changes into the zone of constant temperature at T = T I , as indicated for one- dimension in Figure 4. The fictitious thickness, Delta1x,ofthe wall zone results from approximation (12) in the form of Q = hADelta1T = λA Delta1T Delta1x ,h= λ Delta1x (14) From equation (14), it is possible to estimate the order of magnitude of Delta1x and to compare it with that of the bound- ary layer thickness δ as known from fluid dynamics. The same also applies to influencing the layer thickness and thus h by fluid mechanics measures. Thus, for example, the quantity of h for free convection in air in Table 2 results from the following estimation: assuming a boundary layer for air flows of the order of Delta1x ≈ δ ≈1 mm, a value of h = 0.024/0.001 = 24 W m ?2 K ?1 is obtained from equa- tion (14) with λ air = 0.024 W m ?1 K ?1 (cf. Table 1). How- ever, this value should only be considered as a guide value. Generally, in order to calculate the heat rate by means of equation (12), the heat transfer coefficient h must be known. Therefore, for the representation of the dependence of the heat transfer coefficient, h, on the decisive parame- ters of influence in the practice the dimensionless physical parameters are used. For a large variety of technical configurations, such as tubes, channels, heat exchangers, plates, etc., h has been determined experimentally or theoretically. To reduce the number of empirical equations, the results are usually presented in a dimensionless form. Dimensionless groups can be deduced, which contain the characteristic quantities governing the heat transfer process. These are thermal and fluid properties, characteristic velocities and characteristic length scales. Six dimensionless groups important for the fuel cell technology can be found in the literature (see Refs. [1, 2]) and are mentioned here: the Prandtl number (Pr), the Reynolds number (Re), the Grashof number (Gr), the Nusselt number (Nu), the Schmidt number (Sc), and the Sherwood number (Sh). The definitions are: Re = u ∞ l ν , Gr = gl 3 β(T s ? T f∞ ) ν 2 , Pr = ν α Nu = hl λ , Sc = ν D , Sh = γl D (15) In equations (15), β is the volume coefficient of thermal expansion, u ∞ the free-stream velocity, ν the kinematic viscosity, l the characteristic distance, g the acceleration of gravity and γ the material transfer coefficient. The forced convective flow is described by the Reynolds number, Re. This number represents a measure of the magnitude of the inert forces in the fluid to the viscous forces. Its values indicate either the laminar or the turbulent flow. As an example, for a tube the flow is turbulent for Re 2400. This value can, however, be different for other fuel cell components under consideration. The Nusselt number, Nu, represents the dimensionless heat transfer coefficient h. It is the ratio of the convective heat flux to heat conducted through a fluid layer of the thickness l. Knowledge of this quantity enables the determination of the heat flux through the wall. The convective mass transfer can be analogously treated as the convective heat transfer. In this case a Nusselt number of the second type, the Sherwood number, Sh, is defined and instead of h it describes the mass transfer coefficient γ. Applying the three-fold analogy between momentum, heat and mass transfer, two further groups result. Both members are composed from physical properties only. For the convective heat transfer the Prandtl number, Pr, influ- ences the heat rate. The Prandtl number can be under- stood as the ratio of momentum diffusivity to temperature diffusivity. For air at ambient conditions Pr = 0.7. With increasing Pr the convective heat transfer improves. The same meaning as the Prandtl number for the heat trans- fer, the Schmidt number, Sc, has for the mass transfer. Analogously, it is the ratio of momentum diffusivity mass diffusivity. With increasing Sc, the convective mass transfer improves. In a resting fluid or at low velocities buoyancy 6 Low temperature fuel cells effects may occur, which initiate a free natural convective flow or conjugated natural and forced convection. In this case, the heat transfer depends on the Grashof number, Gr, which is the ratio of the buoyancy forces to inert forces and the square of the friction forces. As mentioned above, the listed groups are suitable for providing the user with information about the heat and mass transfer coefficients. Empirical formulas for the different technical components can be found in the literature. [1–3] Forced convective heat transfer : Nu =f 1 (Re, Pr) Natural convection : Nu =f 2 (Gr, Pr) Forced convective mass transfer : Sh =f 1 (Re, Sc) Natural convective mass transfer : Sh =f 2 (Gr, Sc) The functions f 1 and f 2 can frequently be expressed as simple relationships Nu = aRe n Pr m + b Nu = c(Gr, Pr) p + d(16) with the dimensionless constants a, b, c, d, m, n and p. For the heat transfer equations, the same functions apply as for the mass transfer. In the case of the established laminar flow in tubes and channels, the Nusselt number, Nu, and the Sherwood number, Sh, become independent of the Reynolds and Prandtl numbers. Thus, we have Nu = const. and Sh = const. To set this constant, a value of const. = 4 is a good approximation for the flow channels of a fuel cell. 1.3 Heat transfer with change of state In the LTFCs, processes also take place which