4.4.1. Illustration: determining the dimensions of λ

Solution

Let us recall Fourier’s law:

We know that:

hence the dimensional equation:

Therefore:

4.4.2. Illustration: determining the dimensions of h

Solution

Let us recall the convection equation:

Hence the dimensional equation:

Therefore:

4.5.1. Illustration: dimensions and units of energy

We have seen that energy, as a derived magnitude, admits the dimensional structure given in Table 4.2., which uses the base dimensions, [M], [L] and [t].

Thus, the dimensional equation of energy is written as follows:

Of course, this expression does not depend on the system of units used; it is the same for any system of units, as it is solely linked to the base dimensions and to the structure of the magnitude considered.

We know that energy, on the other hand, may be expressed in joules or in calories, etc.

The following table presents the different energy units which are possible using different standard systems.

4.5.2. Illustration: units of heat conductivity λ

Solution

In section 4.4.1 we saw that the dimensional equation of λ is written as follows:

In order to be able to deduce the units of λ in a coherent system of units, the dimensional groups corresponding to known magnitudes (energy, acceleration, area, temperature, etc.) need to be shown in the dimensional expression of λ.

The structure of the term [M] [t]^-3 [L], which appears in the dimensions of λ, is similar to that of a power. Indeed:

It is therefore possible to identify a power in the dimensional structure of [λ] as follows:

i.e.:

The following table gives the units of λ in different systems.

System of units
IS	W/m °C
Industrial	Cal/s m °C or kcal/hr m °C
Anglo-Saxon	Btu/hr ft °F

The unit kcal/h m °C is the most commonly used in industry.

The tables presented at the end of this book (see Appendix) present conversion factors between the most commonly used systems of units, along with the definitions of certain Anglo-Saxon units.

4.5.3. Illustration: units of the convective transfer coefficient h

Question

From the convective transfer coefficient dimensional equation, determine the units of this coefficient in the following systems: IS, industrial and Anglo-Saxon.

Solution

By conducting a dimensional analysis on the law of convection we can show that the dimensional equation of coefficient h is given by:

Yet, [M][t]^-3 is a power per unit area.

Hence:

The following table gives the units of h in different systems.

System of units
IS	W/m2 °C
Industrial	Cal/s m2 °C or kcal/h m2 °C
Anglo-Saxon	Btu/h ft2 °F

A coherent set of units of h could therefore be J/(s m² °C) or, alternatively, W/(m² °C).

The unit kcal/h m °C is the most commonly used in industry.

The tables given at the end of this book (see Appendix) present conversion factors for the coefficient h between different systems of units.

4.6.1. The Reynolds number

This is defined by:

where:

• d is the inner diameter of the tube, in which the flow occurs.
• v is the velocity of the liquid flowing in the tube.
• ρ designates the density of the flowing fluid.
• μ is the dynamic viscosity of the fluid.

Using the Reynolds number, we can determine the nature of a fluid flow:

• – The flow is said to be “laminar” when Re ≤ 2,300: the fluid flows in parallel layers.
• – The flow is said to be “turbulent” when Re > 4,000: the fluid flows in an irregular manner, which nevertheless leads to an overall defined displacement of the fluid.
• – When 2,300 < Re ≤ 4,000, the flow is said to be “transitional”.

4.6.2. The Nusselt number

This is defined by:

where:

• h is the convective heat transfer coefficient.
• d is the inner or outer diameter of the pipe.
• λ designates the heat conductivity of the flowing fluid.

The Nusselt number may be considered as a measurement of the ratio between the heat flux transferred by convection and that transmitted by conduction.

4.6.3. The Prandtl number

This is defined by:

where:

• Cp is the specific heat of the fluid.
• μ is the viscosity of the fluid.
• λ designates the heat conductivity of the flowing fluid.

We can also write:

Yet:

Hence:

Therefore, the Prandtl number is the ratio of the kinematic viscosity ν (which represents diffusivity for momentum transfer) to thermal diffusivity α.

4.6.4. The Peclet number

This is defined by:

where:

• d is the inner diameter of the tube in which the flow occurs.
• v is the average velocity of the fluid.
• ρ is the density of the fluid.
• Cp is the specific heat of the fluid.
• λ is the heat conductivity of the flowing fluid.

The Peclet number can then be considered as the ratio of the flux of thermal energy transported by the fluid in motion, to the flux of thermal energy transferred by conduction.

It can be noted that:

Hence:

4.6.5. The Grashof number

This is defined by:

where:

• L is the characteristic dimension of the area considered.
• ρ is the density of the fluid.
• g is the acceleration of gravity.
• β is the volumetric expansion coefficient of the constant-pressure fluid.
• ∆ θ is the difference between the wall temperature and the average fluid temperature.
• μ is the viscosity of the fluid.

The Grashof number characterizes the motion of the fluid caused by temperature variations (natural convection). It is analogous to the Reynolds number for forced flow.

4.6.6. The Rayleigh number

This is defined by: Ra = Gr Pr

i.e.:

This is defined by:

where:

• L is the characteristic dimension of the area considered.
• ρ is the density of the fluid.
• g is the acceleration of gravity.
• β is the volumetric expansion coefficient of the constant-pressure fluid.
• ∆ θ is the difference between the wall temperature and the average fluid temperature.
• λ is the heat conductivity of the flowing fluid.
• μ is the viscosity of the fluid.

4.6.7. The Stanton number

where:

• h is the convection heat transfer coefficient.
• C_p is the specific heat of the fluid.
• v is the average velocity of the fluid.
• ρ is the density of the fluid.

It measures the significance of the overall heat flux transferred into the fluid, relative to the heat flux transported by the fluid in motion.

We can easily demonstrate that:

Indeed:

4.6.8. The Graetz number

This is defined by:

where:

• W is the fluid mass flow rate.
• C_p is the specific heat of the fluid.
• λ designates the heat conductivity of the flowing fluid.
• L is the characteristic dimension of the area considered.

It measures the ratio between the heat flux transported by the fluid and the heat flux transferred by conduction.

4.6.9. The Biot number

This is defined by:

where:

• h is the convective heat transfer coefficient between the liquid and the solid.
• s is a characteristic dimension of the solid considered.
• λ designates the heat conductivity of the solid considered.

It gives the relative importance of convection with respect to conduction.

4.6.10. The Fourier number

This is defined by:

where:

• α is the thermal diffusivity of the material considered, defined by
• t designates the time.
• s is a characteristic dimension of the solid considered.

It gives an idea of the rapidity of heat transfer in a solid.

4.6.11. The Elenbaas number

This is defined by:

where:

• L is the length of the base comprising fins.
• z is the distance between two fins.
• ρ is the density of the fluid.
• β is the volumetric expansion coefficient of the constant-pressure fluid.
• g is the acceleration of gravity.
• ∆ θ is the difference between the wall temperature of the fins and the temperature of the fluid at large.
• C_p is the specific heat of the fluid.
• μ is the viscosity of the fluid.

It gives an idea of the importance of natural convection with respect to viscosity forces.

4.6.12. The Froude number

This is defined by:

where:

• v is the average velocity of the fluid.
• g is the acceleration of gravity.
• L is the length of the tube considered.

It gives an idea of the energy transported by a fluid circulating at velocity v, with respect to the friction forces due to gravity over a length L.

4.6.13. The Euler number

This is defined by:

where:

• ΔP is the pressure drop due to the flow of a fluid at velocity v.
• ρ is the density of the fluid.
• v is the average velocity of the fluid.

It gives an idea of the load losses due to flow with respect to the energy transported by a fluid circulating at velocity v.

Table 4.6 presents a summary of dimensionless numbers.

Number type	Expression	Parameter description
Reynolds		d: inner diameter of the tube v: velocity of the fluid ρ: density of the fluid μ: viscosity of the fluid
Nusselt		h: convective heat transfer coefficient d: inner or outer diameter of the tube λ: heat conductivity of the fluid
Prandtl		Cp: specific heat of the fluid μ: viscosity of the fluid λ: heat conductivity of the fluid ν: kinematic viscosity: ν = μ/ρ α: thermal diffusivity: α = λ/ρCp
Peclet		d: inner diameter of the tube v: velocity of the fluid ρ: density of the fluid Cp: specific heat of the fluid λ: heat conductivity of the fluid
Grashof		L: characteristic dimension of the area ρ: density of the fluid g: acceleration of gravity β: volumetric expansion coefficient of the constant-pressure fluid ∆θ: difference between the wall temperature and the average fluid temperature μ: viscosity of the fluid
Stanton		h: convective heat transfer coefficient ρ: density of the fluid Cp: specific heat of the fluid v: velocity of the fluid ρ: density of the fluid
Graetz		W: mass flow rate of the fluid Cp: specific heat of the fluid λ: heat conductivity of the fluid L: characteristic dimension of the area
Biot		h: convective heat transfer coefficient s: characteristic dimension of the solid considered λ: heat conductivity of the solid considered
Fourier		α: thermal diffusivity: α = λ/ρCp λ: heat conductivity of the solid considered ρ: density of the solid considered Cp: specific heat of the solid considered t: time s: characteristic dimension of the solid considered
Elenbaas		L: length of the base comprising fins z: distance between two fins ρ: density of the fluid β: fluid expansion coefficient G: acceleration of gravity ∆θ: Twall - T∞ Cp: specific heat of the fluid μ: viscosity of the fluid.
Froude		v: average velocity of the fluid g: acceleration of gravity L: length of the tube considered
Euler		ΔP: load loss due to the flow of a fluid at velocity v ρ: density of the fluid v: average velocity of the fluid

4.8.1. Illustration: applying Rayleigh’s method

In fluid mechanics, it is demonstrated that the pressure drop ΔP due to the flow of a fluid of viscosity μ in a pipe of inner diameter d depends on the following parameters:

• – The flow velocity of the fluid: v.
• – The density of the fluid: ρ.
• – The viscosity of the fluid: μ.
• – The roughness of the inside walls of the pipe: ε.
• – The pipe length: 1.
• – The pipe diameter: d.
• – The acceleration of gravity: g.

Solutions

1) Basic dimensions of ΔP and of the variables conditioning it.

ΔP has the dimensions of a pressure.

Therefore:

Moreover, the dimensions of the different variables occurring in the definition of ΔP are:

2) Maximum number of equations between parameters

Note that the seven variables and the magnitude concerned (ΔP) are described by just three base dimensions, namely [M], [L] and [t].

Thus, as per Rayleigh, m = 3.

Consequently there are three relations at most that must be satisfied by the exponents involved in the dimensional expression of (ΔP) and the α_j exponents of the seven variables concerned.

3) Dimensional equation

We have:

The dimensional equation is thus written:

or:

Hence:

4) Dimensional equation system

In order to satisfy the previous equation, the parameters (δ₁ … δ₇) need to satisfy the following system of equations:

• – For dimension M: 1 = α₂ + α₃.
• – For dimension L: –1 = α₁ – 3 α₂ – α₃ + α₄ + α₅ + α₆ + α₇.
• – For dimension t: –2 = – α₁ – α₃ – 2 α₇.

Based on this system of equations we can deduce the following relations to be satisfied by the α_i values:

4.8.2. Illustration: verifying Fourier’s law by applying Rayleigh’s method

One of the hypotheses admitted for establishing Fourier’s law is that the variables involved in defining the flux are linked by a simple proportionality. Despite the fact that it admits a justifiable intuitive basis, it is worth verifying this hypothesis.

Thus, in general, it will be assumed that the relation linking the flux ϕ to the

variables and λ is not a simple proportionality, but that it can be put in the following general form:

or, more precisely:

Solutions

1) Basic dimensions of ϕ and of the variables

ϕ has the dimensions of a value of energy per unit time.

Therefore:

i.e.:

or:

Moreover, the dimensions of the different variables occurring in the definition of ϕ are:

2) Maximum number of equations between parameters

Thus, the magnitude ϕ and variables λ, S and are described by four basic dimensions as follows: [M], [L], [t] and [θ].

Therefore, according to Rayleigh’s method, the maximum number m of equations needing to be satisfied by α₁, α₂ and α₃ is m = 4.

3)

The dimensional equation of Fourier’s law is written:

Yet:

Hence:

Thus, the following dimensional system of equations will have to be satisfied:

• – For dimension M: 1 = α₁.
• – For dimension L: 2 = α₁ + 2 α₂ – α₃.
• – For dimension T: –3 = –3 α₁.
• – For dimension θ: 0 = α₃ – α₁.

Or, α₁ = 1

i.e.: α₁ = α₂ = α₃ = 1

4.9.1. Illustration: applying the Buckingham π theorem

Consider again the example presented in section 4.8.1. The sought magnitude G is ΔP. The variables v_i are seven in number; namely:

• – The flow velocity of the fluid: v.
• – The density of the fluid: ρ.
• – The viscosity of the fluid: μ.
• – The roughness of the inside walls of the pipe: ε.
• – The pipe length: l.
• – The pipe diameter: d.
• – The acceleration of gravity: g.

ΔP is written in the form:

Applying Rayleigh's method made it possible to show that the following three relations exist between the α_i parameters.

Solutions

1) Number of independent dimensionless groups

The number m of basic dimensions involved in ΔP and in the parameters likely to be able to describe ΔP is m = 3 (see section 4.8.1 for more details).

Moreover, the number n of variables is n = 7.

By applying the Buckingham πtheorem, we obtain the number p of independent dimensionless groups describing ΔP: p = n + 1 – m.

i.e.: p = 7 + 1 – 3

2) Expression of ΔP

Yet, the following relations are satisfied by the seven α_i parameters:

Hence:

Thus, by replacing α₁, α₂ and α₆ in the expression of ΔP and by rearranging, we obtain:

3) Relation between dimensionless groups

We have:

Hence:

Yet:

Thus, the relation linking ΔP in a dimensionless manner to the groups of parameters conditioning it, can be written in the form:

i.e.:

EXERCISE 4.1. Deriving the convective flux expression by dimensional analysis

We know that the flux transferred by convection between a fluid and a tube through a transfer area A with a thermal potential difference Δθ is conditioned by the following parameters:

• – The driving potential difference Δθ.
• – The transfer area A.
• – The convection heat transfer coefficient h.

In general, we will assume that:

Solutions

1) Maximum number of equations to be satisfied by α1, α₂ and α₃

• – We write:
• – The magnitude G considered is ϕ.
• – The variables are: h, A and Δθ.

According to Rayleigh's method, the maximum number of equations that need to be satisfied by α₁, α₂ and α₃ is equal to the number of basic dimensions involved in the dimensional structures of ϕand variables h, A and Δθ.

From Table 4.2, let us determine the basic dimensions describing ϕ, h, A and Δθ.

We have:

and [Δθ] = [θ]

Thus, the magnitude considered (ϕ) and the variables (h, A and Δθ) are described by the following four base dimensions: [M], [L], [t] and [θ].

Therefore, the maximum number of equations to be satisfied by α₁, α₂ and α₃

is four.

2) Calculating α1, α₂ and α₃

The dimensional equation is written:

or, alternatively, by substituting the dimensions of ϕ and of variables h, A and Δθ:

i.e.:

Hence, by grouping together the terms comprising the same dimensions:

Thus, by identifying the powers relative to each dimension, we obtain:

• – For dimension [M]: 1 = α₁.
• – For dimension [L]: 2 = 2α₂.
• – For dimension [t]: -3 = -3α₁.
• – For dimension [θ]: 0 = α₃ – α₁.

Hence: α₁ = 1; α₂ = 1 and α₃ = α₁ = 1

i.e.:

EXERCISE 4.2. Expressing the radiant flux

We know that a black area S which is at a temperature T emits an energy flux depending on the following parameters:

• – the temperature T.

• – the transfer area A.
• – a constant σ, the dimensions of which are given by

In general, it will be assumed that the flux expression, as a function of these parameters, is of the form:

Solutions

1) Maximum number of equations to be satisfied by α1, α and α3

We have:

In this expression, the magnitude G considered is ϕ, and the variables are: σ, A and T.

According to Rayleigh's method, α₁, α₂ and α₃ must satisfy, at most, a number of equations equal to the number of basic dimensions involved in the dimensional structures of ϕ and of variables σ, A and T.

We have:

Moreover (see Table 4.2), the dimensions describing ϕ, A and T are:

Thus, the magnitude considered (ϕ) and the variables ( σ, A and T) are described

by the four basic dimensions: [M], [L], [t] and [θ].

Therefore, the maximum number of equations that need to be satisfied by α₁, α₂

and α₃ is four.

2) Calculating α1, α₂ and α₃

The dimensional equation is written:

or, alternatively, by substituting the dimensions of ϕ and of variables σ, A and T:

i.e.:

Hence, by grouping together the terms comprising the same dimensions:

Thus, by identifying the powers relative to each dimension, we obtain:

• – For dimension [M]: 1 = α₁.
• – For dimension [L]: 2 = 2α₂.
• – For dimension [t]: -3 = -3α₁.
• – For dimension [θ]: 0 = –4α₁+α₃.

Hence:

i.e.: α₁ = α₂ = 1 and α₃ = 4

EXERCISE 4.3. Driving potential difference in conductive flux

We know that the flux transferred in a solid of heat conductivity λ depends on the following parameters:

• – the area A;
• – the heat conductivity λ and
• – the driving potential difference.

We also know that the flux is given by an equation of the form:

However, we have forgotten whether, in this expression, the driving potential

difference occurs as a difference (Δθ) or as a gradient,

Solutions

1) Dimensions of ϕ, A, λ and Δθ

These dimensions are taken from Table 4.2:

2) Form to be used for the driving potential difference in Fourier's equation

The equation giving the conductive flux is of the form:

The dimensional equation is thus:

Hence:

i.e.:

or:

that is, by simplifying:

It is therefore the dimension of a temperature by a length.

As such, it is the differential that must occur as TPD in the expression of the

conductive flux, rather than Δθ.

EXERCISE 4.4. Obtaining viscosity dimensions from the Grashof number

The Grashof number is given by:

with:

• L, the characteristic dimension of the area considered.
• ρ, the density of the fluid.
• G, the acceleration of gravity.
• β, the volumetric expansion coefficient of the constant-pressure fluid.
• ∆θ, the difference between the wall temperature and the average fluid temperature.
• μ, the viscosity of the fluid.

Solution

The dimensional equation of the Grashof number is written as follows:

Given that Gr is dimensionless, we obtain:

Yet:

and (see Table 4.2):

Hence:

That is, after simplification:

or:

EXERCISE 4.5. Dimensional analysis of the Biot number

By definition, the Biot number is given by:

where:

• h is the convective heat transfer coefficient between the liquid and the solid.
• s is a characteristic dimension of the solid considered.
• λ designates the heat conductivity of the solid considered.

Solution

The dimensional equation of the Biot number gives:

Yet (see Table 4.2):

and: [s] = [L]

Hence:

Therefore, Bi is indeed dimensionless.

EXERCISE 4.6. Dimensional analysis of the Fourrier number

By definition, the Fourrier number is given by:

where:

• α is the thermal diffusivity of the material considered, defined by α = λ/ρΧ_π.
• t designates the time.
• s is a characteristic dimension of the solid considered.

Solution

with:

We have:

i.e.:

Therefore, the dimensional equation of the Fourier number is written:

Yet (see Table 4.2):

and: [s] = [L]

Hence:

It is therefore easy to see that Fo is indeed dimensionless.

EXERCISE 4.7. Dimensional analysis of the Froude number

By definition, the Froude number is given by:

where:

• v is the average velocity of the fluid.
• g is the acceleration of gravity.
• L is the length of the tube considered.

Solution

The dimensional equation relating to the Froude number gives:

Yet (see Table 4.2):

Consequently, Fr is indeed dimensionless.

EXERCISE 4.8. Dimensional analysis of the Euler number

By definition, the Euler number is given by:

where:

• ΔP is the load loss due to the flow of a fluid at velocity v.
• ρ is the density of the fluid.
• v is the average velocity of the fluid.

Solution

The dimensional equation relating to the Euler number gives:

Yet (see Table 4.2):

Hence:

Therefore, Eu is indeed dimensionless.

EXERCISE 4.9. Dimensions of the expansion coefficient β

Let us recall that the Grashof number is defined by:

where:

• L is the characteristic dimension of the area considered.
• ρ is the density of the fluid.
• g is the acceleration of gravity.
• β is the volumetric expansion coefficient of the constant-pressure fluid.
• ∆ θ is the difference between the wall temperature and the average fluid temperature.
• μ is the viscosity of the fluid.

Solution

The dimensional equation relating to the Grashof number gives:

Given that Gr is dimensionless, we obtain:

Yet (see Table 4.2):

Hence:

That is, after simplification:

EXERCISE 4.10. Dimensional analysis of the radiant flux

The flux emitted by a black area S is given by the following equation:

where σ is the Stefan–Boltzmann constant.

Solutions

1) Dimensions of σ

The dimensional equation of the heat flux equation gives:

Hence:

Yet (see Table 4.2):

That is, by substituting in the equation which gives

After simplification, we obtain:

2) The units of σ

We have:

Let us analyze the different groups of dimensions appearing in the dimensional equation of [σ], making the dimensions of some standard magnitudes appear.

[M] [t]-³ recalls the dimensions of an energy flux density or a power per unit surface.

Indeed,

Hence:

Therefore:

σ can then be expressed in W m-² °K-⁴, in kcal hr-¹ m-² °K-⁴, or in Btu hr-¹ ft-² °K-⁴.

EXERCISE 4.11. Dimensional analysis of a mass balance in a mixing tank

The mixing tank represented in Figure 4.3 is used to produce a flow rate D of sugar water, the sugar mass concentration of which must be equal to ρ_D.

For this, a flow E of slightly sweet water (mass concentration ρ_E) is mixed with a liquid sugar concentrate C.

When the flows (E, C and D) and each of the compositions (ρ_E, ρ_C and ρ_D) are given in mass terms, we have demonstrated (see Exercise 1.9) that:

[4.2]

In this exercise, we consider that flows E and D are indeed given in mass terms (kg/hr), but flow C, on the other hand, is given in molar terms (mol/min). Likewise, concentration ρ_E is given in mass terms, whereas concentration ρ_D is given in molar terms.

Solutions

1) Dimensional homogenization

The data for this problem imply the following dimensions for E, D, ρ_E and ρ_D:

• – E is given in mass terms
• – D is given in molar terms
• – ρ_E, ρ_D and ρ_C are fractions (mass or molar); they are therefore dimensionless.

Thus, with this dimensional data for the problem studied, we realize that it is not possible to assure dimensional homogeneity of

Indeed, as the dimensions of E and C are different, the term E + C must be modified as follows, in order to assure its dimensional homogeneity.

– Instead of E + C, we should have E + C M_C, because:

– Instead of C as the numerator, we should have C M_C, because:

Thus, equation [4.2] should be rewritten as follows:

2) Dimensional analysis of equation [4.2]

The dimensional equation of the first term of the equation gives:

Likewise, for the second term we obtain the following dimensions:

Yet:

Hence:

Thus, with the modification applied to equation [4.2], the two terms of this equation will have the same dimensions. Overall dimensional homogeneity is therefore assured.

3) Units proposed

We have:

We propose, for example: min/kg.

EXERCISE 4.12. Dimensional analysis of the mass balance of a reservoir

According to the available for D₁(t) and D₂(t), the differential equation governing the mass balance of the reservoir represented in figure 4.4 can be written in the following three ways:

[4.3]

[4.4]

or

[4.5]

where:

• S is the cross-sectional area of the reservoir.
• ρ is the density of the liquid in the tank.
• C is the molar concentration of the liquid in the tank.
• D₁(t) and D₂(t) are the feed and withdrawal rates. They are given by:
  

Of course, the form (equation [4.3], [4.4] or [4.5]) to be used for the balance equation will depend on the units of D₁(t) and D₂(t).

Indeed, depending on the source of the data, D₁(t) and D₂(t) may be available in:

• a) kg/hr.
• b) l/min or in m³/hr.
• c) mol/min.

Solutions

1) Dimensions of D₁(t) and D₂(t)

• a) Case where flow rates are expressed in kg/hr:

in this case, the dimensions of D₁(t) and D₂(t) are: [D₁] = [D₂] = [M] [t]-¹.

• b) Case where the flow rates are expressed in l/min or in m³/hr:

in this case, the dimensions of D₁(t) and D₂(t) are: [D₁] = [D₂] = [L]³ [t]-¹.

• c) Case where flow rates are expressed in mol/min:

in this case, the dimensions of D₁(t) and D₂(t) are: [D₁] = [D₂] = [mol] [t]-¹.

2) Dimensions and units of coefficients α and β

We have: D₁(t) = α t² and D₂(t) = βte^0,1t

2.1) Dimensions and units of α

The dimensional equation of D₁(t) gives: [D₁] = [α] [t]²

Hence: [α]= [D₁] [t]-²

Table 4.7 gives the dimensions and units of α for each of cases (a), (b) and (c):

Case	Units of D1(t)	Dimensions of D1(t)	Dimensions of α	Units of α
(a)	kg/hr	[M] [t]-1	[M] [t]-3	kg/hr3
(b)	l/min or m3/hr	[L]3 [t]-1	[L]3 [t]-3	l/min3 or (m/hr)3
(c)	mol/min	[mol] [t]-1	[mol] [t]-3	mol/min3

2.2) Dimensions and units of β

The dimensional equation of D₂(t) gives: [D₂] = [β] [t]

Hence: [β]= [D₂] [t]-¹

Table 4.8 gives the dimensions and units of β for each of cases (a), (b) and (c):

Case	Units of D2(t)	Dimensions of D2(t)	Dimensions of β	Units of β
(a)	kg/hr	[M] [t]-1	[M] [t]-2	kg/hr2
(b)	l/min or m3/hr	[L]3 [t]-1	[L]3 [t]-2	l/min2 or m3/hr2
(c)	mol/min	[mol] [t]-1	[mol] [t]-2	mol/min2

3) Form of the equation to be used depending on the case

The three forms are as follows:

– Form (1):

– Form (2):

– Form (3):

The form of the equation to be used will depend on the first member of each of these equations. Dimensional analysis of the first side of each of the forms gives the following results:

– For form (1):

– For form (2):

– For form (3):

Dimensional homogeneity requires these dimensions to be those of D₁(t) and D₂(t).

For each of cases (a), (b) and (c), Table 4.9 gives the dimensions of D₁ and D₂, as well as the form of the equation to be used:

Case	Dimensions of D2(t) and D2(t)	Form to be used
(a)	[M] [t]-1	2
(b)	[L]3 [t]-1	1
(c)	[mol] [t]-1	3

EXERCISE 4.13. Dimensional analysis of the heat balance of a calorimeter

Using the heat balance of a calorimeter, it is possible to link the amount of heat Q that the calorimeter transfers to the fluid, to the increase in temperature (from T₁ to T₂) of the fluid through the following equation:

Where M is the mass of the fluid in the calorimeter and C_p is its heat capacity.

Solutions

1) Dimensions of Q

Q is an energy value. Therefore, its dimensions are (see Table 4.2):

2) Dimensions of C_p

The dimensional equation of the heat balance gives:

Hence:

or:

i.e.:

3) Units of C_p in the International system

We have:

Yet, [L] ² [t]-² is the dimension of an energy per unit mass:

Consequently:

Hence:

In the International System:

Consequently, the units of C_p in the IS are:

4) Units of C_p if Q was in kW-hour

In the present case, we have:

EXERCISE 4.14. Dimensional analysis of the Lennard–Jones potential

The Lennard–Jones potential is used to model the interactions of molecules with one another, rather like in a gravitational field. The potential field ϕ specific to these interactions is given by:

where:

• r is the distance between centers of molecules
• ε is the energy that characterizes the interaction between molecules
• σ is the characteristic diameter of the molecule, also called the collision diameter

Solutions

1) Dimensions of r, ε and σ

2) Dimensions of φ(r)

is dimensionless, therefore:

Thus:

3) Dimensions of F_AB

We have:

Therefore:

Yet:

i.e.:

or:

EXERCISE 4.15. Dimensions and units of the Planck constant

We know that a photon carries an amount of energy E which depends on the wavelength λ(or on the frequency ν) of the radiation according to the relation:

h is the Planck constant.

c is the speed of light.

Solutions

1) Dimensions of E, c and λ

2) Dimensions of h

We have:

Therefore:

Hence:

Thus:

i.e.:

3) Units for h

We know that [M] [L] ² [t]-² is the dimension of an energy.

Thus, in the International System, h will be expressed in Joules × seconds;

i.e.:

EXERCISE 4.16. Dimensional analysis of the heat balance of a water heater

An electric water heater of volume V permits the heating of a flow rate D of water from T₁ to T₂. The heat balance of this heater is given by:

where:

• P is the electrical power of the water heater;
• η is a heating efficiency coefficient;
• V is the volume of the water heater and
• ρ is the density of the water and C_p its sensible heat.

Solutions

1) Dimensions of groups α and β

Dimensional analysis of the differential equation gives:

and:

Hence:

and:

2) Dimensions of η

We have:

Hence:

Yet:

Hence:

Therefore, η is indeed dimensionless.

EXERCISE 4.17. Dimensions of the perfect gas constant

The osmotic pressure πof a solution depends on temperature concentration as follows:

where:

• T is the absolute temperature;
• R is the perfect gas constant;
• V is the volume and a is the solution activity.

Solutions

1) Dimensions of V, T and π

2) Dimensions of R

Thus:

That is, by substituting for [π], [V] and [T]:

i.e.:

3) Units for R

[M] [L] ² [t]-² is the dimensional structure of an energy.

Thus, R may be expressed in J/°C, in cal/°C, in Btu/°F, etc.

EXERCISE 4.18. Dimensions of the osmosis coefficient

During the balancing by osmosis of the concentrations, C₁ and C₂, of two compartments, the osmotic flux density is expressed as follows:

φ is the molar density of the osmotic flux.

C₁ and C₂ are the solute molar concentrations in compartments 1 and 2, respectively.

E is the thickness of the semi-permeable membrane.

D is the solute diffusion coefficient through osmosis; this data is provided by the manufacturer of the semipermeable membrane.

Solutions

1) Dimensions of φ, C and E

• φ is a density of the molar flux → [φ] = [mol] [t]-¹ [L]-²
• C is a molar concentration → [C] = [mol] [L] -³
• E is a thickness → [E] = [L]

2) Dimensions of D

Hence:

That is, by substituting the dimensions of φ, C and E:

i.e.:

3) Units for D

cm²/sec, m²/min, in²/sec, etc.

EXERCISE 4.19. Dimensions of a heat balance equation

The stirring and heating tank presented in Figure 4.5 enables sweet juice to be produced by mixing a flow d of sugar syrup with a flow D of pure water.

In order to accelerate dissolution, the tank is stirred with a propeller mixer and heated by steam.

The heat balance in steady state is written as follows:

The different parameters are defined in the technical data sheet presented in Table 4.10.

Solutions

1) Dimensions of Q, V, D and J and of the sensible heats

i.e.:

V, D, d and J are mass flow rates → [V] = [M] [t]-¹

The sensitive heat values have the dimensions of an energy per unit mass and per unit temperature.

i.e.:

or

2) Dimensions of Λ

The heat balance equation is written as:

Dimensional homogeneity implies: [V Λ] = [Q]

3) Units of Λ

We have: [Λ] = [L]² [t]-²

We know that [M] [L] ² [t]-² is the dimensional structure of an energy.

Therefore, [L] ² [t]-² is the dimensional structure of an energy per unit mass.

As a result, Λ may be expressed in J/g, in cal/kg, in Btu/lb, etc.

EXERCISE 4.20. Homogeneity of the reverse-osmosis sizing equation

In a multi-stage reverse-osmosis system (see Figure 4.6), the total number N of modules needed to perform a given desalination operation is linked to the mass flow rate ϕ to be treated by:

where:

• ϕ is the total mass flow rate;

  the flow rate treated by each module is then:

• η is the number of stages in the system;
• m is the number of reverse-osmosis modules per stage;
• φ_s is the mass flux density that can be reached by a module;
• σ is the transfer area of a cell: manufacturer's data.

Solutions

1) Dimensions of φ, ϕ_S and σ

2) Verifying the dimensional homogeneity

We have:

Because N is dimensionless, homogeneity will require:

With n being dimensionless, we need to have:

This is satisfied because:

EXERCISE 4.21. Permeability dimensions of a reverse-osmosis membrane

During a reverse-osmosis purification operation, the flux density of the solvent passing through the semipermeable membrane, under the action of a pressure gradient ΔP, is given by:

where:

• φ_S is the mass flux density of the solvent passing through the semi-permeable membrane;
• ψ_S is the membrane's permeability to the solvent;
• ΔP is the difference in pressure between the two compartments;
• E is the thickness of the membrane.

Solutions

1) Dimensions of φS , ΔP and E

φ_S is a mass flux density

ΔP has the dimensions of a pressure value

E is a thickness → [E] = [L]

2) Dimensions of ψΣ

We have:

Hence:

That is, by substituting in the dimensions of φ_Σ, ΔP and E:

or:

3) Units for ψ_S

or:

or: , etc.

EXERCISE 4.22. Dimensions of the overall dialysis coefficient

The flux of a solute passing through the membrane of a dialyzer is given by the following equation (Lane & Riggle, 1959):

where:

• ϕ is the mass flux of the solute passing through the membrane.
• U is the overall dialysis coefficient.
• C is data provided by the membrane manufacturer.
• A is the transfer area: membrane area.
• ΔC_mL is the logarithmic mean of the difference in concentration between the two sides of the membrane:
  

Solutions

1) Dimensions of ϕ, ΔC_mL and A

ΔC_mL has the dimensions of a mass concentration.

Indeed:

Hence:

i.e.:

A is an area → [A] = [L]²

2) Dimensions of the overall dialysis coefficient U

The overall dialysis coefficient is linked to the mass flux by:

Hence:

That is, by substituting the dimensions:

i.e.:

3) Units for U

U can therefore be expressed in cm/s, in ft/min, etc.

EXERCISE 4.23. Permeability dimensions of a dialysis membrane

In dialysis, an important parameter is the permeability U_m of the membrane to the solute. The following empirical relation links this parameter to the diffusion coefficient D of this solute through the membrane (dimensions: [D] = [L]² [t]-¹):

where:

• F is a non-dimensional factor.
• ε is the volume fraction of the membrane occupied by the pores.
• h is the tortuosity, defined as the ratio between the capillary length of the pores and the membrane thickness.
• z is the wet thickness of the membrane.

Solution

U_m is given by:

Hence:

As F, ε and h are dimensionless, then:

Hence:

i.e.:

EXERCISE 4.24. Dimensions of the separation capacity of a centrifuge

The separation capacity, ΔU_max, of a Zippe centrifuge is expressed as a function of its peripheral velocity V as follows:

where:

• Z is the centrifuge height;
• C is the molar concentration;
• D is the diffusion coefficient (dimensions: [D ] = [L]² [t]-¹);
• ΔM is the difference between the molar masses of the two molecules to be separated;
• V is the peripheral velocity of the centrifuge;
• T is the absolute temperature;
• R is the perfect gas constant.

Solutions

1) Dimensions of C, ΔM, V and R

C is a molar concentration

ΔM has the dimensions of a molar mass

V is a velocity

By definition, _R is given by:

and: [Pressure] = [M] [L]-¹ [t]-²

Hence:

Therefore:

2) Dimensions of the ratio

The dimensional equation is thus:

i.e. by substituting the dimensions of the different magnitudes:

χ is then dimensionless.

3) Dimensions of the overall dialysis coefficient U

We have

Hence:

or:

That is, by replacing by the dimensions:

That is, ultimately:

4) Units of U:

U can then be expressed in mol/sec.

EXERCISE 4.25. Dimensions of flux transferred by electrodialysis

In an electrodialysis unit, the flux density of the mass transferred depends on the electric field imposed at the cell terminals. The relation that reflects this dependency is not dimensionless. It gives the max flux density as a function of the parameters of the problem:

where:

• z_i is the valence of the ion considered;
• v is the average velocity (cm/s);
• ∇E is the electric field gradient (volts/cm);
• F is the Faraday constant (C/g);
• c_i is the concentration of i (g/cm₃);
• λ_ι is the ion conductance (cm₂/Ω g);
• u_i is the ion mobility of i:

Solutions

1) Dimensions of φι, v, ∇ Ε, z_i, F and c_i.

Yet (see Table 4.2): [E] = [M] [L]² [t]-³ [I]-¹

Hence: [∇E] = [M] [L] [t]-³ [I]-¹

2) Dimensions of u₁

Dimensional homogeneity results in:

Hence:

3) Dimensions of λi

We have:

The dimensional equation of this expression of u_i gives:

or, alternatively, by replacing [u_i] with

Hence:

That is, by substituting the dimensions of [φ_ι], [F], [c_i] and [∇E]:

That is, after simplification:

i.e.:

λ_i is the ion conductance → [λ_ι] = [Area] [Electrical resistance]-¹[M] -¹

Yet: [Electrical resistance] = [M] [L]² [t]-³ [I]-²

Hence:

i.e.:

4) Dimensions of D_i

Dimensional homogeneity results in:

Hence:

i.e.:

or:

5) Units of φi

We have:

The units of φ_iwill be those of the term c_iv:

Hence: <φ_i> = g/cm³sec.

EXERCISE 4.26. Dimensions of electrodialysis power consumption

The power consumption of an electrodialyzer is given by the equation:

with:

• E, the average specific consumption (kWh/m³);
• F, the water volumetric flow rate (m³/hr);
• R_e, the electrical resistance of a cell of the electrolyzer (ohms);
• A_e, the active area of the membrane of a cell of the electrolyzer (m²);
• U, the voltage applied to the terminals of an electrolysis cell (volts);
• n, the number of cells.

Questions

• 1) Determine the dimensions of E.
• 2) As a function of the units given below, what units of E would result from the equation?

Data:

Solutions

1) Dimensions of E

Yet:

• F is a volumetric flow rate → [F] = [L]³ [t]-¹
• R_e is an electrical resistance → [R_e] = [M] [L]² [t]-³ [I]-²
• A_e is an area → [A_e] = [L]²
• U is an electric potential difference → [U] = [M] [L]² [t]-³ [I]-¹

Hence:

or:

i.e.:

2) Units of E

We know that [M] [L] ² [t]-² is the dimensional structure of an energy.

Thus:

Therefore:

4.11.1. Osborne Reynolds

Renowned Irish physicist Osborne Reynolds was born on August 23, 1842 in Belfast (Northern Ireland).

With a background in engineering, Reynolds made important contributions that helped to structure fluid dynamics theory.

He is thus considered to be one of the founders of fluid mechanics, thanks to his many contributions to hydrodynamics in general, and to fluid dynamics in particular.

His most significant contribution to the study of fluids in motion was the establishment of a methodology to characterize a fluid's motion through the calculation of a dimensionless number. This number, which as we know is of great importance to the practice of fluid engineering, came to be known as the Reynolds number.

Indeed, it was Reynolds who, in 1883, first proposed using this number to determine whether water flow will occur in parallel (laminar) layers or rather in a sinuous manner (turbulent). This theoretical development, tried and tested through experimentation, appeared in January 1883 in his article entitled “An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels”, published in Philosophical Transactions, volume 174 (Royal Society of London).

As well as being a brilliant researcher, Osborne Reynolds also taught engineering. He was among those who argued that all engineering students should have a solid foundation of knowledge in mathematics and physics. Despite his great interest in teaching he was not, however, a great teacher! In the opinion of his former students, his classes were complicated and hard to follow. He had difficulty simplifying the knowledge he had to convey: a great scientist but a poor educator, often changing subject during a presentation without taking care to ensure the necessary connections and transitions.

Nevertheless, and thanks above all to his brilliant contributions to the field of research, he became the holder of one of only two engineering chairs existing at that time. He directed the chair of the unit, which later became the University of Victoria in Manchester. He died on February 21, 1912 in Watchet (England).

4.11.2 Ludwig Prandtl

When characterizing Ludwig Prandtl, one would be quick to describe him as a highly-valued scientist.

His training was both conventional and solid. After obtaining a degree in physics from Ludwig-Maximilian University in Munich, he began studying for a postgraduate degree in engineering at the same university.

His dissertation, presented on November 14, 1899, entitled Lateral Displacement Phenomena, a Case of Unstable Equilibrium, dealt with the stability of combustion equilibria. He continued his research at the same university, where he obtained the title of Doctor of Science.

He began his working life developing assembly lines for MAN factories. Here, the realities on the ground first led him to develop an interest in fluid mechanics. Indeed, it was as a result of analyzing the operating mode of a dust collector that he came to explore the fundamentals and explanations that could lead to potential improvements in performance.

He taught at the University of Hanover (Gottfried-Wilhelm Leibnitz University) and at the University of Göttingen where, betwee en 1906 and 1908, he supervised the research work of the young Theodore von Kármán with the goal of obtaining his doctorate.

Ludwig Prandtl's contributions to boundary layer theory, which he had introduced in 1904, offered him unchallenged leadership in the field. He was appointed director of the brand-new Max Planck Institute for Fluid Dynamics in Göttingen in 1909.

Starting in 1907, he focused on supersonic flows and the shock waves that must accompany them.

The prospects that opened up as a result of nascent aerodynamics and aeronautics applications gave great importance to his work. He then developed a method for visualizing the oscillations of air layers at the outlet of a de Laval nozzle and in 1908, he oversaw the construction of Germany's first wind tunnel.

In 1910, he was the first to introduce a dimensionless number into the description of turbulent flows. This number came to be called the Prandtl number.

Analysis of supersonic flows and the experimental installations at his disposal allowed him to then develop his theory of airfoils, which would come to form the basis of aerodynamics and modern aeronautical engineering. Thus, he developed an acceptable formula for the calculation of lift, and developed, back in 1919, a drag theory enabling the design and calculation of aircraft wing profiles. In 1929, he developed a reactor calculation method that continues to be used in missile design.

Four major achievements of this great man of science are to be remembered in particular, since they remain to this day of great use in engineering calculations and in aerodynamic technologies:

• 1) Boundary layer theory, which makes it possible to explain several manifestations of mass diffusion heat transfer.
• 2) The dimensionless number, which he introduced in 1910 to describe turbulent flows. This number, which subsequently became known as the Prandtl number, continues to be used in transfer calculations.
• 3) His work on hydraulics, published in 1931 under the title Führer durch die Strömungslehre.
• 4) The probe he developed for determining the relative velocity of an object in an aerodynamic environment. This is the Prandtl probe.

Prandtl's work published in 1931 can be considered as the ultimate reference for studies on laminar and turbulent flow, boundary layer, fluid mechanics, supersonic flows and the shock waves that must accompany them, as well as aerodynamics and aeronautical applications. This work also serves as a reference in the field of visualization of air-layer oscillations at the outlet of a nozzle.

The Prandtl probe, a variant of the Pitot tube, continues to be used in aerodynamics to this day: using this probe we are at present able to appreciate an aircraft’s velocity in relation to the wind.