Engineering Principles and Transport Processes

Ideal gas law applications

The ideal gas law PV = nRT links pressure, volume, temperature and the number of moles n, with R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}. Because n = m/M, a measured mass and volume at known P and T give the molar mass M, which is the usual route to identifying an unknown gas. A relation written in another form, such as PV = cRT with a numerical constant c, is read by comparing it term by term with PV = nRT, so c is the number of moles.

When two distinct ideal gases sit in separate vessels that are later connected, each gas behaves independently. Each keeps its own n_i, so P_iV_i = n_iRT holds before and after the connection, where P_i and V_i are the initial pressure and volume of gas i. At constant temperature the final partial pressure of gas i is P_iV_i/V_{\text{total}}, and the equilibrium pressure is the sum of these partial pressures, as stated by Dalton's law.

For a process in an insulated closed system against a constant external pressure, no heat is exchanged, so q = 0 and \Delta U = -W, with W = P_{\text{ext}}\Delta V. For an ideal gas \Delta U = nC_V\Delta T, which gives the final temperature. Expansion against an external pressure lowers the internal energy and therefore cools the gas.

For a reversible isobaric process in a closed system, the heat transferred equals the enthalpy change, Q = \Delta H = nC_p\Delta T. For an ideal gas C_p = C_V + R on a molar basis, so the heat capacities must be used on the same basis as n.

Heat transfer coefficient

The heat transfer coefficient h (W m^{-2} K^{-1}) relates the heat flux at a surface to the temperature difference across the film: q = hA\Delta T. Its reciprocal is the film resistance per unit area. When a wall separates two fluids, the film resistances of the two fluids and the conduction resistance of the wall act in series and are added.

For a thin wall of thickness x and conductivity k, the overall coefficient U satisfies \frac{1}{U} = \frac{1}{h_i} + \frac{x}{k} + \frac{1}{h_o}. The overall coefficient is always smaller than the smaller of the individual coefficients, because every resistance adds to the total.

The overall coefficient links the total duty to the heat transfer area through Q = UA\Delta T_m, where \Delta T_m is the appropriate mean driving force. When one fluid condenses at a constant saturation temperature, the driving force changes only through the temperature rise of the other fluid, so the log-mean difference between the terminal temperature differences is used. For a tube of inside diameter d and length L, the inside area is A = \pi dL, which allows the length to be found from the area.

For a body cooling in a surrounding at constant temperature T_\infty, Newton's law of cooling states \frac{dT}{dt} = -k(T - T_\infty). Integration gives an exponential approach to T_\infty, so T - T_\infty = (T_0 - T_\infty)e^{-kt}. Two temperature readings at known times are enough to determine k and then to predict the temperature at any later time.

It is also useful to remember that some expressions in these calculations reduce to a limit in which numerator and denominator both vanish. For \lim_{x\to 0}\frac{x - \sin ax}{x - \sin bx} with a, b \neq 1, the result is \frac{1-a}{1-b}, which follows from the small-angle approximation \sin ax \approx ax or from L'Hôpital's rule.

Clausius-Clapeyron equation

The Clapeyron equation gives the slope of a phase boundary on a P–T diagram: \frac{dP}{dT} = \frac{\Delta H}{T\,\Delta V}. For solid–liquid equilibrium, \Delta H_{\text{fus}} is the molar enthalpy of fusion and \Delta V_{\text{fus}} is the molar volume of the liquid minus that of the solid. Rearranging gives the latent heat from a measured slope: \Delta H_{\text{fus}} = T\,\Delta V_{\text{fus}}\left(\frac{dP}{dT}\right).

For liquid–vapour equilibrium, when the vapour behaves as an ideal gas and the liquid volume is negligible, the equation becomes the Clausius-Clapeyron form \frac{d\ln P}{dT} = \frac{\Delta H_{\text{vap}}}{RT^2}, which assumes that \Delta H is nearly constant over the temperature range.

• molar volume change — the difference in molar volume between the two phases; its sign decides the slope direction.
• unit consistency — pressure in Pa and volume in m^3 mol^{-1} give enthalpy in J mol^{-1}.

For water, which expands on freezing, \Delta V_{\text{fus}} is negative, so the melting line has a negative slope.

Heat exchanger LMTD

In a double-pipe exchanger the heat duty is Q = UA\Delta T_{lm}, and the heat flux per unit area is q/A = U\Delta T_{lm}. The log-mean temperature difference is \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}, where \Delta T_1 and \Delta T_2 are the temperature differences between the two fluids at the two ends of the exchanger. The order of the two terms does not matter, but the pair must be taken consistently. When \Delta T_1 = \Delta T_2, the LMTD equals that common value.

The pairing of terminal differences depends on the flow arrangement. In counter-current flow the outlet of the cold fluid meets the inlet of the hot fluid, so the pairs are hot inlet with cold outlet and hot outlet with cold inlet. In co-current flow the pairs are inlet with inlet and outlet with outlet. For the same terminal temperatures, counter-current flow gives the larger LMTD, which is why it is preferred. Differences in kelvin and degrees Celsius are numerically equal, so either unit can be used for \Delta T.

Stokes settling velocity

Stokes' law gives the terminal settling velocity of a small rigid sphere in a viscous fluid at low particle Reynolds number, typically below about 1: v_t = \frac{\Delta\rho\, g\, d^2}{18\mu}. Here \Delta\rho is the density difference between particle and liquid, d is the particle diameter and \mu is the dynamic viscosity. Because the velocity varies with the square of the diameter, doubling the diameter quadruples the settling velocity.

If a particle is assumed to reach its terminal velocity at once, the time to settle through a tank of height H is t = H/v_t. Use SI units throughout, with \mu in Pa s and \Delta\rho in kg m^{-3}, so that v_t comes out in m s^{-1}. Outside the creeping-flow range, the drag coefficient must be used instead of the Stokes expression.

Material balance with recycle

The degrees of freedom of a process are the number of variables that can be specified independently, given by \text{DOF} = N_{\text{variables}} - N_{\text{independent equations}}. The variables are the flow rates and compositions of the streams. For a stream containing C components, the mole fractions carry C-1 independent variables once the sum constraint is applied, so each stream adds one flow rate plus C-1 composition variables.