Thermal Energy Equation Term - Chain Rule

In summary: The Attempt at a SolutionOne of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule... @kevman90 Do you know how to take the derivative of ## \frac{d(uv)}{dt}##? It is ## u (\frac{dv}{dt}) +v(\frac{du}{dt}) ##. In this case, ## u=P ## and ## v=1/\rho ##. With the chain rule, ## \frac{dv}{dt}=(\frac{dv}{d \rho}) (\frac{d \r
  • #1
kevman90
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2

Homework Statement


I am going through a derivation of the thermal energy equation for a fluid and am stumped on one of the steps. Specifically, the text I am using converts the term:

P/ρ*(Dρ/Dt)

to:

ρ*D/Dt(P/ρ) - DP/Dt

where:
ρ = density
P = pressure
D/Dt = material derivative

The text says this is done using the chain rule of differentiation but I can't derive it myself. I'm far removed from calculus so maybe I'm missing something simple but any help would be appreciated.

Homework Equations

The Attempt at a Solution

 
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  • #2
One of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule... @kevman90 Do you know how to take the derivative of ## \frac{d(uv)}{dt}##? It is ## u (\frac{dv}{dt}) +v(\frac{du}{dt}) ##. In this case, ## u=P ## and ## v=1/\rho ##. With the chain rule, ## \frac{dv}{dt}=(\frac{dv}{d \rho}) (\frac{d \rho}{dt}) ##. Do you know how to compute ## \frac{d v}{d \rho} ## ? With that, you should be able to process the second expression that you have, but I think you will find that it equals the minus of your first expression.
 
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  • #3
Charles Link said:
One of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule...
Sorry @Charles Link -- I was in the process of deleting the OP and warning for not showing enough work. But if you want to give a couple hints, that's probably okay.
 
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This makes sense - didn't think about using the product rule. I will work through it later but I think I've got it. Also my mistake with the minus sign I forgot to include it out in front of the first term. Thanks!
 
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What is the thermal energy equation term?

The thermal energy equation term is a mathematical representation of the amount of thermal energy present in a system. It takes into account factors such as temperature, heat capacity, and mass to quantify the thermal energy in a given system.

What is the chain rule in relation to thermal energy equations?

The chain rule is a calculus principle that is used to find the derivative of a function that is composed of two or more sub-functions. In the context of thermal energy equations, the chain rule is used to calculate the change in thermal energy with respect to a specific variable, such as temperature or time.

Why is the chain rule important in thermal energy equations?

The chain rule is important in thermal energy equations because it allows us to find the rate of change of thermal energy with respect to a specific variable. This is crucial in understanding how thermal energy behaves in a system and how it is affected by various factors.

How is the chain rule applied in thermal energy equations?

The chain rule is applied in thermal energy equations by taking the derivative of each individual sub-function and then multiplying them together. This allows us to find the overall rate of change of thermal energy with respect to a specific variable.

What are some practical applications of the thermal energy equation term - chain rule?

The thermal energy equation term - chain rule is used in various fields such as engineering, physics, and chemistry to analyze and predict thermal energy behavior in systems. It is also used in the development of thermal management systems and in the study of thermodynamics and heat transfer.

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