Intermediate chemical engineering: 50 equations and principles

All 50 cards

1. 1
   Steady-state energy balance on an open system
   $\dot{Q} - \dot{W}_s = \Delta\dot{H} + \Delta\dot{E}_k + \Delta\dot{E}_p$

   $\dot{Q}$ is heat added to the system, $\dot{W}_s$ shaft work done by it, and $\Delta$ means outlet minus inlet. The kinetic and potential energy terms are often negligible.
2. 2
   Enthalpy change for heating without a phase change
   $\Delta\hat{H} = \displaystyle\int_{T_1}^{T_2} C_p(T)\,dT$

   Exact for ideal gases and close for liquids and solids. With constant $C_p$, $\Delta\hat{H} = C_p\,\Delta T$.
3. 3
   Heat of vaporization of water at 100 °C
   About $2257\ \mathrm{kJ/kg}$, or $40.7\ \mathrm{kJ/mol}$

   A latent heat: it is absorbed at constant temperature and pressure while the phase changes.
4. 4
   Standard heat of reaction from heats of formation
   $\Delta H_r^\circ = \displaystyle\sum_i \nu_i\,\Delta H_{f,i}^\circ$

   $\nu_i$ is positive for products and negative for reactants. The heat of formation of an element in its standard state is zero.
5. 5
   Mechanical energy balance
   $\dfrac{\Delta P}{\rho} + \dfrac{\Delta v^2}{2} + g\,\Delta z + \hat{F} = -\dfrac{\dot{W}_s}{\dot{m}}$

   For an incompressible fluid at steady state; $\hat{F}$ is the friction loss per unit mass. With no friction or shaft work it becomes Bernoulli’s equation.
6. 6
   Why use recycle?
   To send unreacted feed back to the reactor, raising overall conversion and cutting raw material use and waste

   Recycle also recovers solvents and catalysts.
7. 7
   Single-pass vs overall conversion
   Single-pass: the fraction of reactant entering the reactor that reacts in it. Overall: the fraction of reactant fed to the whole process that reacts

   With unreacted feed recycled, overall conversion can be far higher than single-pass conversion.
8. 8
   Purge stream
   A small stream drawn off a recycle loop to stop inert gases or impurities from building up

   Example: argon and methane purged from the ammonia synthesis loop. Some reactant is lost with them.
9. 9
   Bypass stream
   Part of a feed that skips a process unit and rejoins its outlet, used to control the final composition or temperature
10. 10
    Raoult’s law
    $y_i P = x_i P_i^{\text{sat}}(T)$

    For ideal liquid solutions with an ideal vapor; most accurate when a component is nearly pure ($x_i$ close to 1).
11. 11
    Henry’s law
    $y_i P = x_i H_i(T)$

    For a dilute solute ($x_i$ close to 0), such as a sparingly soluble gas. $H_i$ is the Henry’s law constant, in pressure units in this form.
12. 12
    Antoine equation
    $\log_{10} P^{\text{sat}} = A - \dfrac{B}{T + C}$

    An empirical vapor pressure fit. $A$, $B$ and $C$ are tabulated for each substance, for specific units and a temperature range.
13. 13
    Relative volatility
    $\alpha_{ij} = \dfrac{y_i/x_i}{y_j/x_j}$

    Under Raoult’s law, $\alpha_{ij} = P_i^{\text{sat}}/P_j^{\text{sat}}$. The closer it is to 1, the harder the separation by distillation.
14. 14
    Bubble-point temperature
    The temperature at which a liquid heated at constant pressure forms its first bubble of vapor

    For an ideal solution, solve $\sum_i x_i P_i^{\text{sat}}(T) = P$ for $T$.
15. 15
    Dew-point temperature
    The temperature at which a vapor cooled at constant pressure forms its first drop of liquid

    For an ideal solution, solve $\sum_i \dfrac{y_i P}{P_i^{\text{sat}}(T)} = 1$ for $T$.
16. 16
    Azeotrope
    A mixture whose equilibrium vapor has the same composition as the liquid, so ordinary distillation can’t separate it further

    Ethanol and water form one at about 95.6 wt% ethanol at 1 atm.
17. 17
    Reflux ratio
    $R = \dfrac{L}{D}$: the liquid returned to the column per unit of distillate withdrawn
18. 18
    Effect of raising the reflux ratio
    Fewer stages are needed for the same separation, but reboiler and condenser duties, and so energy costs, rise

    The optimum balances the column’s capital cost against its energy cost.
19. 19
    Constant molar overflow
    The McCabe–Thiele assumption that the liquid and vapor molar flows are constant within each section of the column

    It holds when the components have similar molar heats of vaporization and heat losses are small.
20. 20
    Rectifying-section operating line
    $y_{n+1} = \dfrac{R}{R + 1}\,x_n + \dfrac{x_D}{R + 1}$

    It crosses the diagonal $y = x$ at $x_D$, the distillate composition.
21. 21
    Feed line (q-line)
    $y = \dfrac{q}{q - 1}\,x - \dfrac{z_F}{q - 1}$

    For a two-phase feed, $q$ is its liquid fraction. A saturated liquid feed ($q = 1$) gives a vertical line; a saturated vapor ($q = 0$), a horizontal one.
22. 22
    Counting stages in a McCabe–Thiele diagram
    Step between the equilibrium curve and the operating lines from one end of the column to the other; each step is one equilibrium stage

    A partial reboiler counts as one equilibrium stage.
23. 23
    Gas absorption vs stripping
    Absorption transfers a component from a gas into a liquid solvent; stripping (desorption) transfers it from a liquid into a gas

    Example: removing $\mathrm{CO_2}$ from natural gas with an amine solution.
24. 24
    Absorption factor
    $A = \dfrac{L}{mV}$

    $L$ and $V$ are the liquid and gas molar flows, $m$ the slope of the equilibrium line. With $A > 1$, enough stages can absorb nearly all of the solute.
25. 25
    Liquid–liquid extraction
    Moves a solute from one liquid into an immiscible solvent in which it is more soluble

    The solvent-rich product is the extract; the treated feed left behind is the raffinate.
26. 26
    Distribution coefficient in extraction
    $K_D = \dfrac{y}{x}$ at equilibrium

    $y$ is the solute concentration in the extract phase and $x$ in the raffinate phase. A larger $K_D$ means less solvent is needed.
27. 27
    Reynolds number for pipe flow
    $Re = \dfrac{\rho v D}{\mu}$

    Flow is laminar below about 2100, turbulent above about 4000, and transitional in between.
28. 28
    Darcy–Weisbach equation
    $h_f = f_D\,\dfrac{L}{D}\,\dfrac{v^2}{2g}$

    $f_D$ is the Darcy friction factor, read from the Moody chart or the Colebrook equation. The Fanning factor is $f_F = f_D/4$.
29. 29
    Friction factor in laminar pipe flow
    $f_D = \dfrac{64}{Re}$, or in Fanning form $f_F = \dfrac{16}{Re}$

    In laminar flow it doesn’t depend on pipe roughness.
30. 30
    Pump operating point
    Where the pump curve (head delivered vs flow) crosses the system curve (head the piping needs vs flow)

    Closing a valve steepens the system curve and moves the operating point to a lower flow.
31. 31
    Net positive suction head (NPSH)
    The NPSH available at the pump suction must exceed the NPSH the pump requires, or the liquid boils and the pump cavitates

    NPSH available is the absolute suction head above the liquid’s vapor pressure head. Cavitation erodes the impeller.
32. 32
    Pump affinity laws for a change in speed
    $Q \propto N$, $\;H \propto N^2$, $\;P \propto N^3$

    Flow, head and power against rotational speed $N$, for the same impeller.
33. 33
    Minor loss through a valve or fitting
    $h_L = K\,\dfrac{v^2}{2g}$

    $K$ is the fitting’s loss coefficient. Losses can also be given as an equivalent length of straight pipe.
34. 34
    Log mean temperature difference (LMTD)
    $\Delta T_{\text{lm}} = \dfrac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$

    $\Delta T_1$ and $\Delta T_2$ are the hot-to-cold temperature differences at the two ends of the exchanger.
35. 35
    Heat exchanger design equation
    $Q = UA\,\Delta T_{\text{lm}}$

    For multipass or crossflow exchangers, multiply $\Delta T_{\text{lm}}$ by a correction factor $F \le 1$.
36. 36
    Overall heat transfer coefficient for a thin flat wall
    $\dfrac{1}{U} = \dfrac{1}{h_i} + \dfrac{\Delta x}{k_w} + \dfrac{1}{h_o}$

    Resistances in series: inside film, wall (thickness $\Delta x$, conductivity $k_w$) and outside film. For tubes, each term is scaled by its area.
37. 37
    Fouling
    Deposits such as scale, corrosion products or biological growth on heat transfer surfaces, which add thermal resistance and lower U

    Designers add fouling resistances $R_f$ to $1/U$, which makes the exchanger larger.
38. 38
    Power-law rate law and reaction order
    $-r_A = k\,C_A^{\alpha} C_B^{\beta}$; the overall order is $\alpha + \beta$

    Orders are found by experiment; for an elementary reaction they equal the stoichiometric coefficients.
39. 39
    Arrhenius equation
    $k = A\,e^{-E_a/RT}$

    $A$ is the pre-exponential factor and $E_a$ the activation energy. A plot of $\ln k$ against $1/T$ has slope $-E_a/R$.
40. 40
    Integrated first-order rate law at constant volume
    $\ln\dfrac{C_{A0}}{C_A} = kt$

    The half-life, $t_{1/2} = \dfrac{\ln 2}{k}$, doesn’t depend on the starting concentration.
41. 41
    Units of the rate constant for an nth-order reaction
    $(\text{concentration})^{1-n}\,(\text{time})^{-1}$

    First order: $\mathrm{s^{-1}}$. Second order: $\mathrm{L\,mol^{-1}\,s^{-1}}$.
42. 42
    Batch reactor design equation
    $N_{A0}\,\dfrac{dX}{dt} = -r_A V$

    $N_{A0}$ is the initial moles of A and $X$ its conversion. Integrating gives the time to reach a given conversion.
43. 43
    CSTR design equation
    $V = \dfrac{F_{A0}\,X}{-r_A}$

    The rate is evaluated at the exit conditions, since a perfectly mixed tank has the exit composition throughout.
44. 44
    PFR design equation
    $V = F_{A0}\displaystyle\int_0^X \dfrac{dX}{-r_A}$

    From the differential form $F_{A0}\,\dfrac{dX}{dV} = -r_A$.
45. 45
    Space time
    $\tau = \dfrac{V}{v_0}$

    Reactor volume divided by the entering volumetric flow rate $v_0$. It equals the mean residence time when the volumetric flow is constant.
46. 46
    CSTR vs PFR volume for a positive-order reaction
    A PFR needs less volume for the same conversion, because a CSTR runs entirely at the low exit concentration

    Several CSTRs in series approach the volume of a PFR.
47. 47
    Feedback control
    Measure the controlled variable, compare it with the set point, and adjust a manipulated variable to reduce the error

    It can only correct a disturbance after it has affected the process.
48. 48
    PID control law
    $$u(t) = \bar{u} + K_c\left[e(t) + \frac{1}{\tau_I}\int_0^t e(t')\,dt' + \tau_D\,\frac{de}{dt}\right]$$

    $e$ is the error (set point minus measurement), $K_c$ the controller gain, $\tau_I$ the integral (reset) time, $\tau_D$ the derivative time and $\bar{u}$ the bias.
49. 49
    Offset in proportional-only control
    A steady-state error that remains after a set-point change or sustained disturbance; integral action removes it

    A higher proportional gain shrinks the offset but can make the loop oscillate.
50. 50
    Controlled, manipulated and disturbance variables
    Controlled: what is held at a set point. Manipulated: what the controller adjusts. Disturbance: an input that is not adjusted but upsets the process

    Example: a tank’s temperature is controlled by adjusting steam flow, while the feed temperature is a disturbance.