Cantera/Reactor Equations - Revision history

Admin at 08:49, 17 April 2017

2017-04-17T08:49:00Z

← Older revision		Revision as of 08:49, 17 April 2017
Line 586:		Line 586:


			=Flags=

	~~[[Category:Cantera]]~~		{{CanteraFlag}}

Admin: /* 1D Model */

2014-03-14T21:28:57Z

1D Model

← Older revision		Revision as of 21:28, 14 March 2014
Line 495:		Line 495:
	</math>		</math>

	the time is non-dimensionalized as		<math>\alpha_{ji}</math> is the stoichiometric coefficient of species i for reaction j, the time is non-dimensionalized as

	<math>		<math>
Line 501:		Line 501:
	</math>		</math>

	and a is the scaling constant for distance, <math>\phi^2_j</math> is the Thiele modulus~~, and~~		and a is the scaling constant for distance, <math>\phi^2_j</math> is the Thiele modulus.

	~~<math>~~
	~~\beta_j = \dfrac{ ( -\Delta H_j ) D c_f }{ k T_f }~~
	~~</math>~~

	===1D Energy Balance===		===1D Energy Balance===
Line 519:		Line 515:
	<math>		<math>
	\mathbb{L} = \frac{ k \varepsilon }{ D C_p }		\mathbb{L} = \frac{ k \varepsilon }{ D C_p }
			</math>

			beta relates to release of energy via reaction,

			<math>
			\beta_j = \dfrac{ ( -\Delta H_j ) D c_f }{ k T_f }
	</math>		</math>

Admin: /* Equations from Froment and Bischoff */

2014-02-21T00:29:27Z

Equations from Froment and Bischoff

← Older revision		Revision as of 00:29, 21 February 2014
Line 524:		Line 524:

	=Equations from Froment and Bischoff=		=Equations from Froment and Bischoff=

			The following are (mostly) equivalent reactor equations from Froment and Bischoff's book ''Chemical Reactor Analysis and Design'' (1990).

	==1D Model==		==1D Model==

Admin: /* Equations from Aris */

2014-02-21T00:28:33Z

Equations from Aris

← Older revision		Revision as of 00:28, 21 February 2014
Line 468:		Line 468:

	=Equations from Aris=		=Equations from Aris=

			The following are (mostly) equivalent reactor equations from Aris's book ''The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts'' (1975).

	==1D Model==		==1D Model==

Admin: /* Units */

2014-02-21T00:26:54Z

Units

← Older revision		Revision as of 00:26, 21 February 2014
Line 463:		Line 463:
	A word on units of important quantities.		A word on units of important quantities.

	Volume units are <math>\text{m}^3</math>		Volume units are <math>\text{m}^3</math>, area units are <math>\text{m}^2</math>

			Site density is <math>\frac{ \text{sites} }{ \text{m}^2 }</math>

	=Equations from Aris=		=Equations from Aris=

Admin: /* Surface Coverage Equation */

2014-02-21T00:25:37Z

Surface Coverage Equation

@@ Line 342: / Line 342: @@
 See also: [[Cantera/Surface Coverage]]
 In order to model surface reactions, Cantera treats them similar to the way it treats gas phase reactions. In a gas phase reaction, the reaction rate of a simple reaction like:
@@ Line 384: / Line 399: @@
 The surface coverage, then, is a "mole fraction" of a species on the surface. The surface coverages also evolve with time, due to changing reaction conditions and production rates, so the surface coverages must be evolved in time using a differential equation, just like the species.

Admin: /* Species Equations (Gas) */

2014-02-21T00:23:42Z

Species Equations (Gas)

← Older revision		Revision as of 00:23, 21 February 2014
Line 245:		Line 245:
	* Mass flow into reactor		* Mass flow into reactor
	* Mass flow out of reactor		* Mass flow out of reactor

			In words: the species are changed due to gas-phase (volumetric) reactions, surface-phase (surface) reactions, and net convection into/out of the reactor.

	The net production rate of species k can be further broken down as:		The net production rate of species k can be further broken down as:

Admin: /* Energy Equation */

2014-02-21T00:22:50Z

Energy Equation

← Older revision		Revision as of 00:22, 21 February 2014
Line 67:		Line 67:

	where <math>h_w</math> is the heat transfer coefficient for the wall, <math>A_w</math> is the wall area, <math>\Delta T</math> is the temperature difference across the wall, <math>\varepsilon_w</math> is the emissivity of the wall, <math>\sigma</math> is the Stefan constant, <math>G_w(t)</math> is an arbitrary, user-specified, time-dependent energy source term, <math>\dot{m}_i</math> is the mass flowrate of all streams entering into the reactor, <math>h_{i,in}</math> is the enthalpy of the streams entering into the reactor, and <math>h</math> is the enthalpy of any stream leaving the reactor.		where <math>h_w</math> is the heat transfer coefficient for the wall, <math>A_w</math> is the wall area, <math>\Delta T</math> is the temperature difference across the wall, <math>\varepsilon_w</math> is the emissivity of the wall, <math>\sigma</math> is the Stefan constant, <math>G_w(t)</math> is an arbitrary, user-specified, time-dependent energy source term, <math>\dot{m}_i</math> is the mass flowrate of all streams entering into the reactor, <math>h_{i,in}</math> is the enthalpy of the streams entering into the reactor, and <math>h</math> is the enthalpy of any stream leaving the reactor.

			In words: the energy change is due to compression work, heat transfer to the system through walls in the reactor, heat transfer via wall radiation/emissivity, and net enthalpy being convected into/out of the reactor.

	Note that this is a more specific form of the general energy balance derived from [[Reynolds Transport Theorem]]:		Note that this is a more specific form of the general energy balance derived from [[Reynolds Transport Theorem]]:

Admin: /* Volume Equation */

2014-02-21T00:21:31Z

Volume Equation

← Older revision		Revision as of 00:21, 21 February 2014
Line 12:		Line 12:

	where <math>K_w</math> is a wall factor that controls the dependence of wall velocity on pressure gradient across the wall, <math>A_w</math> is the wall area, <math>\Delta P</math> is the pressure difference across the wall (so that the wall velocity is pressure-dependent), and <math>F_w(t)</math> is an arbitrary, user-specified, time-dependent volumetric source term.		where <math>K_w</math> is a wall factor that controls the dependence of wall velocity on pressure gradient across the wall, <math>A_w</math> is the wall area, <math>\Delta P</math> is the pressure difference across the wall (so that the wall velocity is pressure-dependent), and <math>F_w(t)</math> is an arbitrary, user-specified, time-dependent volumetric source term.

			In words: the volume change is due to expansion of all walls connected to our reactor, plus a user-defined volume change function.

	===Volume Equation Code===		===Volume Equation Code===

Admin: /* 1D Model */

2014-01-28T22:30:39Z

1D Model

← Older revision		Revision as of 22:30, 28 January 2014
Line 460:		Line 460:

	Aris gives non-dimensional forms of the mass and energy balances in a catalyst pellet as follows:		Aris gives non-dimensional forms of the mass and energy balances in a catalyst pellet as follows:

			===1D Material Balance===

	For the material balance,		For the material balance,
Line 490:		Line 492:
	\beta_j = \dfrac{ ( -\Delta H_j ) D c_f }{ k T_f }		\beta_j = \dfrac{ ( -\Delta H_j ) D c_f }{ k T_f }
	</math>		</math>

			===1D Energy Balance===

	For the energy balance,		For the energy balance,