I am currently working on my bachelor's thesis and I am trying to derive Bernoulli's equation which can be used for changing temperatures. Let's say I got a pipe with inflow-area $A_1$ and outflow-area $A_2$. I also know the heat I am feeding into the pipe $\dot{Q}$. Furthermore, the inflow velocity and inflow pressure are known. I also know the pressure loss from friction $\Delta p_r$
I am going to assume that the density of my fluid will not change a lot. This means that I can get the outflow velocity using:
$$v_2 = v_1 \cdot \dfrac{A_2}{A_1}$$
Applying the first law of thermodynamics:
$$
0 = \dot{Q}+\sum_i \dot{m}_i \left( h_i+g\cdot z_i+\frac{c_i^2}{2}\right)
$$
Since I want to get an equation similar to Bernoulli's equation with an additional temperature term, I would first split $h_i$ into something depending on the pressure and the temperature.
$$h_i = c_p\cdot T+p\cdot?$$
I am very unsure of how this should be formulated and even derived. I would end up plugging this one back into the first law of thermodynamics and ideally end up with the Bernoulli equation with a temperature term.
I am very happy for any help or advice!
Best Answer
Bernoulli's equation focuses only on mechanical energy while the first law focuses on every kind of energy (including mechanical). Also, unless you specially include the effects of viscous drag in Bernoulli'e equation, it will not properly nclude all the different kinds of mechanical energy. Once the mechanical energy balance equation is properly expressed, it can be subtracted from the first law equation (overall energy balance) to yield a new equation which might be called the "thermal energy balance equation" (which includes temperature changes and viscous heating). See Bird, Stewart, and Lightfoot, Transport Phenomena, for more details.