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Bioengineering

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    Overview - types of mass

    The amount of a material is expressed through the base physical variables of mass or mole. Mass \( m \) is a quantity of matter that has weight in a gravitational field. Common units are grams \( g \), kilograms \( kg \), pounds \( lb_m \), etc.

    Extra!

    Callout Card

    Note the \( m \) on \( lb_m \)! This indicates that it represents the mass in Imperial Units within Earth's gravitational field. Later on, we will define \( lb_f \) a force unit in Imperial Units, so it's important to include the \( m \) when using this unit.

    The mole \( n \) is a base unit describing an amount of any substance containing Avogadro’s number of molecules of that substance. One mole contains \( 6.02x10^{23} \) atoms of that element and has a mass, in grams, equal to the atomic weight of the element. For example, a single molecule of O\textsubscript2 has an atomic weight of 32.0 amu; in one mole of O\textsubscript2, there are \( 6.02x10^{23} \)molecules of O\textsubscript2, having a mass of 32.0 g collectively.

    The molecular weight, \( M \), of component A is related to the mass, \( m \), and the number of moles, \( n \), of that component:

    $$ n_A = \frac{m_A}{M_A} $$

    Common units are g/mol and lbm/lbm-mol.

    Mass flow rate describes the transport of material over a period of time.

    $$ \dot{m} = Av\rho $$

    Where A is the cross-sectional area of a cylindrical tube.

    $$ A = \frac{\pi}{4} D^2 = \pi r^2 $$

    Conservation statements

    The following equations can be written for any unit of mass including total mass, species mass, element mass, total moles, element moles, species moles. If the system is a non-compressible gas, then volume can be used.

    Algebraic Equation

    $$ \sum{m_{in}}-\sum{m_{out}}+\sum{m_{gen}}-\sum{m_{cons}} = m_{acc} $$

    Differential Equation

    $$ \sum{\dot{m}_{in}}-\sum{\dot{m}_{out}}+\sum{\dot{m}_{gen}}-\sum{\dot{m}_{cons}} = \frac{m^{sys}}{dt} $$

    Integral Equations

    $$ \int^{t_f}_{t_0}\sum{\dot{m}_{in}}-\int^{t_f}_{t_0}\sum{\dot{m}_{out}}+\int^{t_f}_{t_0}\sum{\dot{m}_{gen}}-\int^{t_f}_{t_0}\sum{\dot{m}_{cons}} =\int^{t_f}_{t_0} \frac{m^{sys}}{dt} $$

    Multicomponent mixtures

    Consider systems with mass entering and leaving in streams. Each chemical species or compound, s, in each stream is associated with its species flow rate as \( \dot{n}_s \) (moles of s/time) or \( \dot{m}_s \) (mass of s/time]). The total flow rate of the stream, in either moles or mass, can be calculated by summing the individual species flows over all species s present in the stream:

    $$ \dot{m}=\sum_{s}{\dot{m}_s} $$
    $$ \dot{n}=\sum_{s}{\dot{n}_s} $$

    An alternative way of representing a stream is to give its total flow, in rate of either moles or mass, together with the composition of the stream. Two convenient measures of composition of a species are the mass or weight fraction, \( w_s \), and the mole fraction, \( x_s \). All mass and mole fractions of all species s in a stream must sum to 1:

    $$ 1=\sum_{s}{w_s} $$
    $$ 1=\sum_{s}{x_s} $$

    Where mass and mole fractions are related to mass and molar flow rates, respectively

    $$ w_s=\frac{\dot{m}_s}{\dot{m}} $$
    $$ x_s=\frac{\dot{n}_s}{\dot{n}} $$

    and using the molecular weight according to

    $$ \dot{n}_s=\frac{\dot{m}_s}{M_s} $$

    The differential conservation equations for systems containing multiple inlets and outlets can be extended to systems that contain multiple species in each stream. For open, nonreacting, steady-state systems, the following mass conservation equations are written for:

    Species mass: \( \sum{\dot{m}_{in,s}}-\sum{\dot{m}_{out,s}}=0 \)

    Species moles: \( \sum{\dot{n}_{in,s}}-\sum{\dot{n}_{out,s}}=0 \)

    Element mass: \( \sum{\dot{m}_{in,e}}-\sum{\dot{m}_{out,e}}=0 \)

    Element moles: \( \sum{\dot{n}_{in,e}}-\sum{\dot{n}_{out,e}}=0 \)

    Systems with chemical reactions

    When approaching a problem involving a chemical reaction, you must write out the reaction and balance the equation before moving any further into the problem. Stoichiometric balances and reaction rates must always be worked in units of moles or molecules.

    The reaction rate (R) characterizes the extent to which a chemical reaction proceeds. Reaction rate is expressed in moles or moles/time.

    R is a constant for a stoichiometric equation and is not tied to a specific species and/or compound in a reacting system. R may be given, deduced, or calculated using the following:

    $$ \sum{\dot{n}_{in,s}}-\sum{\dot{n}_{out,s}}+\sigma_sR=0 $$

    or

    $$ R = \frac{\dot{n}_{in,s}-\dot{n}_{out,s}}{-\sigma_s} $$

    The fractional conversion (f) of a reactant is the fraction of reactant s that reacts in the system relative to the total amount of s introduced into the system.

    $$ f_s=\frac{\dot{n}_{cons, s}}{\dot{n}_{in, s}}=\frac{\dot{n}_{in,s}-\dot{n}_{out,s}}{-\dot{n}_{in,s}}=\frac{-R\sigma_s}{\dot{n}_{in, s}} $$

    Finally, the limiting reactant is mathematically defined as the minimum of the following expression for all species in the system.

    $$ \bigl\{\frac{\dot{n}_{in, s}}{-\sigma_s}\bigr\} $$