Clarifying the Definition Total/Stagnation Pressure (p₀) in Bernoulli'

[John1704]

Homework Statement

Question From Paper: A hot air balloon has an envelope which has height h = 30 m high (see below). The balloon is always operated with a temperature difference between the heated gas and the atmosphere of 100K. For simplicity, assume that the envelope contains heated air only (that is, ignore combustion products).

With reference to the figure, when points C and D are at an altitude of 2 km, calculate the static and total pressures at A, B, C and D. Describe how these pressures lead to lift.

Relevant Equations

Bernoulli's Equation

Question From Paper: A hot air balloon has an envelope which has height h = 30 m high (see below). The balloon is always operated with a temperature difference between the heated gas and the atmosphere of 100K. For simplicity, assume that the envelope contains heated air only (that is, ignore combustion products).

With reference to the figure, when points C and D are at an altitude of 2 km, calculate the static and total pressures at A, B, C and D. Describe how these pressures lead to lift.

Hot Air Balloon Schematic

My Question:

I have trouble understanding what exactly is "total pressure". In Bernoulli's Equation, total pressure seems to be the sum of the static, dynamic, and gravitational potential term.
If I apply that definition, I will come to the conclusion that total pressure at A, B, C, and D is equal, while static pressure varies depending on location.

Assuming relative velocity of air, v, = 0 at all points since the balloon moves with the same velocity as the surrounding air; and also take the reference datum of Z = 0 at C and D, we get:

Ptotal,C = Pstatic,C + ρg(0)

Ptotal,D = Pstatic,D + ρg(0)

Since C and D are both exposed to the atmosphere, Pstatic,C = Pstatic,D and so Ptotal,C = Ptotal,D.

Considering point A inside the balloon,

Pstatic,A = Pstatic,C - ρhot airgh

Similarly,

Pstatic,B = Pstatic,D - outside airgh

Hence,

Ptotal,A = Pstatic,C - ρhot airgh + ρhot airgh

And

Ptotal,B = Pstatic,D - ρoutside airgh + ρoutside airgh

The last bolded terms are the GPE terms in Bernoulli's equation)

Hence I get the conclusion that total pressure at A, B, C, and D are the same.

The solution seems to use a different definition of total pressure which disregards the GPE per volume term and hence come to the conclusion that the total pressure at points A, B, C and D are different.

Does my line of reasoning make sense? Or is the solution correct?

Based on the formula of total pressure, it seems that the value of total pressure will vary depending on where we define the reference datum since the ρgZ term in Bernoulli's equation depends on the height from the datum. Is this true?
Thank you very much for reading and helping with my confusion.

 

[gmax137]

Paging @Chestermiller

 

[Chestermiller]

Homework Statement: Question From Paper: A hot air balloon has an envelope which has height h = 30 m high (see below). The balloon is always operated with a temperature difference between the heated gas and the atmosphere of 100K. For simplicity, assume that the envelope contains heated air only (that is, ignore combustion products).

With reference to the figure, when points C and D are at an altitude of 2 km, calculate the static and total pressures at A, B, C and D. Describe how these pressures lead to lift.
Relevant Equations: Bernoulli's Equation

Question From Paper: A hot air balloon has an envelope which has height h = 30 m high (see below). The balloon is always operated with a temperature difference between the heated gas and the atmosphere of 100K. For simplicity, assume that the envelope contains heated air only (that is, ignore combustion products).

With reference to the figure, when points C and D are at an altitude of 2 km, calculate the static and total pressures at A, B, C and D. Describe how these pressures lead to lift.

Hot Air Balloon Schematic

My Question:

I have trouble understanding what exactly is "total pressure". In Bernoulli's Equation, total pressure seems to be the sum of the static, dynamic, and gravitational potential term.
If I apply that definition, I will come to the conclusion that total pressure at A, B, C, and D is equal, while static pressure varies depending on location.

Assuming relative velocity of air, v, = 0 at all points since the balloon moves with the same velocity as the surrounding air; and also take the reference datum of Z = 0 at C and D, we get:

Ptotal,C = Pstatic,C + ρg(0)

Ptotal,D = Pstatic,D + ρg(0)

Since C and D are both exposed to the atmosphere, Pstatic,C = Pstatic,D and so Ptotal,C = Ptotal,D.

Considering point A inside the balloon,

Pstatic,A = Pstatic,C - ρhot airgh

Similarly,

Pstatic,B = Pstatic,D - outside airgh

Hence,

Ptotal,A = Pstatic,C - ρhot airgh + ρhot airgh

And

Ptotal,B = Pstatic,D - ρoutside airgh + ρoutside airgh

The last bolded terms are the GPE terms in Bernoulli's equation)

Hence I get the conclusion that total pressure at A, B, C, and D are the same.

The solution seems to use a different definition of total pressure which disregards the GPE per volume term and hence come to the conclusion that the total pressure at points A, B, C and D are different.

Does my line of reasoning make sense? Or is the solution correct?

Based on the formula of total pressure, it seems that the value of total pressure will vary depending on where we define the reference datum since the ρgZ term in Bernoulli's equation depends on the height from the datum. Is this true?
Thank you very much for reading and helping with my confusion.

View attachment 359992

Bernoulli equation: $$p+\frac{1}{2}\rho v^2+\rho g z=Const.$$

p="static pressure"

$\frac{1}{2}\rho v^2$ = "dynamic pressure"

$p+\frac{1}{2}\rho v^2$ = "total pressure"

It seems to me your assessment was correct. The gravitational term is not included in what they conventionally define as the total pressure. I might add that I have a very low regard for use of this type of terminology, but, of course, you need to know it since many people employ it.