Isochoric and isobaric heat of a reacton

Eicosane · June 2, 2023, 03:47

Hello everyone!

I am trying to incorporate chemical reaction calculation in my compressible FVM solver, and there is a question occured about how to account heat of reaction.

In most examples, where such effect is considered, energy equation is solved regarding enthalpy of a flow. But i use in my solver the HLLC scheme to calculate flows, thus for energy balance equation total energy is used. So i'm trying to find a way to calculate heat effect of reaction in terms of total or internal energy and not enthalpy.

In chemical literature i found two variants of a heat effect of a reaction. Isochoric of reaction at constant volume and isobaric at constant pressure. They are expressed as
$Q_{V} = \Delta U^{298}_{V=const}$
$Q_{p} = \Delta H^{298}_{p=const}$
For ideal gas there relation is
$\Delta H^{}_{p=const} - \Delta U^{}_{V=const} = \Delta n \cdot R T$
But the $Q_{V}$ never described any further and never used in the texts, only $Q_{p}$ is, as reactions in laboratory chemistry mostly considered isobaric.

Additionaly to my mind comes the thought, that solving equation of chemical reaction in finite volume cells, we can consider each cell independently as a fixed volume reactor of ideal mixing, so reactions that flow in cell are actually not isobaric, but isochoric.

So my question is, how should i calculate energy change due to chemical reaction?
$U^{n+1} = U^{n} + \Delta U^{298}_{V=const} \cdot \Delta \xi$
or
$H^{n+1} = H^{n} + \Delta H^{298}_{p=const} \cdot \Delta \xi$
or maybe
$U^{n+1} = U^{n} + \Delta H^{298}_{p=const} \cdot \Delta \xi$
where $\Delta \xi$ -- calculated number of reactions per volume on n step,

,

-- internal energy and enthalpy per volume correspondingly.

LuckyTran · June 2, 2023, 08:02

Lab experiments are most often isobaric because they mix their chemicals in a petri dish that's open to ambient. Using a combustion example, an isochoric reaction is what happens when you burn in a piston-cylinder engine with no stroke, i.e. a bomb.

Assuming your energy equation is indeed working properly except for the source term and has no bugs, consider that if you use the isochoric heat of reaction then you should use the constant volume specific heat (cv) in your formulation of total energy. If you use isobaric heat of reaction, then use constant pressure specific heat (cp). Now that would be an equation in Temperature which you are not interested in because you say you want internal energy, but the transport equations that you write down using all these different approaches must be consistent with one another. The actual flow process undergone by the fluid will be solved by listening to the transport equation, not our emotions. Real flows are neither isochoric nor isobaric. Isochoric heats and constant volume specific heats are both unpopular because most flows are not inside rigid tanks with closed ends and having to keep track of the boundary work that would occur all the time is tedious. Of course you can still do it (in your code for example) if you are willing to deal with the extra bookkeeping.

Hint: you need to write down the entire energy transport equation and look at where the pressure work term is.

Eicosane · June 5, 2023, 14:37

Quote:

Originally Posted by LuckyTran

Hint: you need to write down the entire energy transport equation and look at where the pressure work term is.

Thank you for the reply.

I use division on processes in my work, so there is several energy equations, that are solved.
On the advection stage i'm solving Euler equation for total energy, using HLLC:

$\frac{\partial (\varrho E)}{\partial t} + \nabla \cdot (\varrho E \cdot \vec{u}) + \nabla \cdot (p \cdot \vec{u}) = 0.$

Next i consider various diffusive effects. There i substract kinetic energy from total and get internal energy, which express in terms of temperature:

$\frac{\partial (\varrho \cdot \bar{C}_{V} \cdot T)}{\partial t} - \nabla \cdot (\frac{\bar{C}_{V}\cdot \mu}{Pr} \cdot \vec\nabla T) + \nabla \cdot (\sum_{k} h_{k}\cdot \vec{j^{D}_{k}}) = \sigma : \vec{\nabla}\vec{u}.$

And for the chemical reaction stage, i guess, i can solve equation:

$\frac{\partial U}{\partial t} = \dot{Q}$

where:

$\dot{Q} = \Delta U_{form} \cdot \frac{\partial \xi}{\partial t}$

Is this approach plausible?

June 2, 2023, 03:47	Isochoric and isobaric heat of a reacton	#1
Eicosane New Member Maksim Join Date: Jul 2020 Posts: 10 Rep Power: 6	Hello everyone! I am trying to incorporate chemical reaction calculation in my compressible FVM solver, and there is a question occured about how to account heat of reaction. In most examples, where such effect is considered, energy equation is solved regarding enthalpy of a flow. But i use in my solver the HLLC scheme to calculate flows, thus for energy balance equation total energy is used. So i'm trying to find a way to calculate heat effect of reaction in terms of total or internal energy and not enthalpy. In chemical literature i found two variants of a heat effect of a reaction. Isochoric of reaction at constant volume and isobaric at constant pressure. They are expressed as $Q_{V} = \Delta U^{298}_{V=const}$ $Q_{p} = \Delta H^{298}_{p=const}$ For ideal gas there relation is $\Delta H^{}_{p=const} - \Delta U^{}_{V=const} = \Delta n \cdot R T$ But the $Q_{V}$ never described any further and never used in the texts, only $Q_{p}$ is, as reactions in laboratory chemistry mostly considered isobaric. Additionaly to my mind comes the thought, that solving equation of chemical reaction in finite volume cells, we can consider each cell independently as a fixed volume reactor of ideal mixing, so reactions that flow in cell are actually not isobaric, but isochoric. So my question is, how should i calculate energy change due to chemical reaction? $U^{n+1} = U^{n} + \Delta U^{298}_{V=const} \cdot \Delta \xi$ or $H^{n+1} = H^{n} + \Delta H^{298}_{p=const} \cdot \Delta \xi$ or maybe $U^{n+1} = U^{n} + \Delta H^{298}_{p=const} \cdot \Delta \xi$ where $\Delta \xi$ -- calculated number of reactions per volume on n step, , -- internal energy and enthalpy per volume correspondingly.

June 2, 2023, 08:02		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,728 Rep Power: 66	Lab experiments are most often isobaric because they mix their chemicals in a petri dish that's open to ambient. Using a combustion example, an isochoric reaction is what happens when you burn in a piston-cylinder engine with no stroke, i.e. a bomb. Assuming your energy equation is indeed working properly except for the source term and has no bugs, consider that if you use the isochoric heat of reaction then you should use the constant volume specific heat (cv) in your formulation of total energy. If you use isobaric heat of reaction, then use constant pressure specific heat (cp). Now that would be an equation in Temperature which you are not interested in because you say you want internal energy, but the transport equations that you write down using all these different approaches must be consistent with one another. The actual flow process undergone by the fluid will be solved by listening to the transport equation, not our emotions. Real flows are neither isochoric nor isobaric. Isochoric heats and constant volume specific heats are both unpopular because most flows are not inside rigid tanks with closed ends and having to keep track of the boundary work that would occur all the time is tedious. Of course you can still do it (in your code for example) if you are willing to deal with the extra bookkeeping. Hint: you need to write down the entire energy transport equation and look at where the pressure work term is. Last edited by LuckyTran; June 2, 2023 at 10:39.

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