Q&A forum: Centrifugal and Axial Fan Calculator

Your client has raised a number of issues regarding fan design;
conflicting opinions, efficiency, turbulence, noise, temperature, forward-facing blade configurations, etc.
all of which I will try to deal with separately.

1) First of all, the Fans calculator only calculates the performance of the impeller (or squirrel cage), not the fan. It was intended to take the drudgery (repetitive calculation) out of the design process.

2) Fans calculates air-flow through the impeller as defined by Charles Innes; nothing is added or taken away from his theory.

3) Despite his theory being developed over a hundred years ago, it remains today the most reliable and predictable theory on this subject. And whilst more than 95% of the world's fan designers use it in one form or another, most tend to modify their calculations with preferred factors and coefficients, which is the reason discrepancies and conflicts of opinion appear. Ask 20 designers how to achieve best results for given conditions, and you will get 10 different answers (see 11) below).

4) You can calculate efficiencies any way you wish; friction, shock, inlet vs outlet; pressures, velocities, energies, heads, etc., each of which will vary differently with blade configuration; as one goes up, other(s) may go down (and vice versa). Fans only calculates those efficiencies defined by Charles Innes. So, before you specify the relevances of efficiencies or inefficiencies for any blade configuration, you need to be sure as to which you are referring.

5) An important thing to remember is that there should be no turbulence in a properly designed impeller. Charles Innes theory quite correctly assumes this to be the case. You will only generate turbulence if you disrupt air-flow by; skewing and/or curving your blades, spacing them too closely, alter air-flow direction more than once, employ forward-facing blades with inappropriate inlet blade angles, etc. In all such cases the calculation results achieved will be unreliable.

6) As your client correctly states, the shoulder of one blade should not be inside the radial line of the toe of its neighbour. However, you should carry out 7) below for best results.

7) A useful tip for blade-spacing is to calculate an optimum blade configuration and then draw the inlet and outlet flow diagrams (as shown on the calculator and our technical help web page; Fig 3) using actual angles and velocity scales, and use vᵢ and vₒ to define blade spacing.

8) Most single-stage fans generally result in a minimal increase in pressure and therefore have little effect in the impeller's temperature related efficiency. You can work this out using the δpVRT formula. You will probably find that the resultant temperature rise will be just a few degrees Kelvin at most (<1%).

9) Because turbulence should be zero, any noise referred to (on our technical help page) is related entirely to the 'whoosh' that would be heard due to air-flow. The greater the air-flow the louder the whoosh (noise). If you are getting noise from turbulence, your blades are not working according to Charles' theory.

10) With regard to aspect ratio for preferred output results, this is best carried out by calculation with variable input data and optimising for best efficiency. Remember, 'energy-in' always equals 'energy-out'. Best results are not always achieved with short blades or long blades. You should always optimise aspect ratios and impeller width through calculation.

11) Your client is also correct with regard to the use of forward-facing blades. Their benefits are questionable.
In reality, as demonstrated in the Accuracy section at the bottom of the technical help page for this calculator - as long as the vₒ-vᵢ conflict is not significant - you can obtain similar results with backward and forward-facing blades, according to Charles Innes. But impeller performance can be seriously affected by small (inappropriate) changes in toe-angles of forward-facing blades. In practice, properly designed forward-facing blades will increase air-pressure within the impeller and reduce airflow.

12) CalQlata’s rules of thumb are just that; general guidelines. They should not be taken literally.

Question: "A longer chord enlarges the widening in the gap been adjacent blades. I assume a widening in the gap between blades slows the flow and increases the pressure (similar to the centrifugal shroud)." However, when I increase the chord length in the Fans software of a forward facing fan, the delta pressure figure stays the same. Why is this?
Answer: If the blade spacing is correct - i.e. there is no unpressurised air-space - as the flow of air is via a combination of centrifugal force, annular pressure and frictional drive, it should be the same for all air passing between the blades, irrespective of chord length; you simply need to optimise blade spacing.

Question: "When setting blade outlet angles greater than 90°, always set the inlet blade angle shallow enough to overcome inward thrust from the outlet tip”. Are you saying that the blade’s airfoil action is opposing the outward flow and the centrifugal force is overwhelming this?
Answer: Yes; this is what is meant by the statement; vₒ & vᵢ are working against each other.

To summarise, your client should assume that he is getting no turbulence in his impeller as calculated by ‘Fans’, and apply preferred factors and coefficients to his input data if he wishes.
I (personally) prefer to rely on a clean (unmodified) analysis (no turbulence) and use backward facing blades where possible to avoid air-flow (vₒ & vᵢ) conflicts, with minimum losses. I also prefer to alter fan characteristics via the casing if possible.

"In answer to your client's question, the simple answer is yes (but that is not what it was designed for).
However, he/she must enter correct values for: pᵢ, pₒ, Rₐ, Cᶠ, γ
Rₐ = SHC : water is 4645.754757538 J/K/kg # (internet value: 4183)
Cᶠ is likely to be greater than 0.3 for water on steel
γ is the specific heat ratio (pressue:volume) ≈1 (4.65:4.51) #
All of the above are suggestions only, your customer should use his/her preferred values. But calculation accuracy will be dependent upon accuracy of the data entered.
# calculated; not as defined in the internet"

Moreover, efficiency values are likely to be lower than those calculated for a gas, and foreward facing blades are not recommended for liquids.
(You should bear in mind that Charles Innes' theory was originally created for a compressible fluid (air), whereas water is virtually incompressible. However, because air passing through a fan is not significantly compressed, calculation accuracy is not expected to be significant.)

(post-reply comments)

The calculation data you refer to in Q&A (bottom of this page) is that of a customer. It is not CalQlata's data. And the data is erroneous; θₒ = 120°.
The latest version of the calculator compares well with proprietary items as can be seen in the example calculation at the bottom of the technical help page for the this calculator.

The earlier version of this calculator used an optimisation factor to take into account the number of blades. The latest version of the calculator offers a blade-count as an input.
If we recalculate the example in Q&A using the latest version of the calculator entering an optimum blade-count of 13, we get Q = 17.582584, which is similar to the Q&A result of Q = 17.2368 (0.004788 x 3600).
But also bear in mind that the earlier version of this calculator did not deal with incorrect input data so well.

Fans is simply a theoretical calculator designed to take the drudgery out of impeller calculations for fan (casing and equipment) designers. It has most probably become CalQlata's most verified calculator due to the number of customers that appear to have difficulty with the theoretical concept and therefore question its output.

As stated in the technical help page for this calculator, Fans calculates the movement of air through an impeller (not through a fan) according to Charles Innes, which may or may not be better than any other related theory. However, whilst his theory has been around for more than a hundred years and is well known in the industry, as we all know, there is more than one way to 'skin-a-cat'. I suggest therefore that if your client already has access to software with which he is happy, he may be well advised to continue with it.

Charles Innes' theory dealt with air-flow only and does not address torque (and therefore power). CalQlata has incorporated its own torque formulas in Fans, which have been used for many years and appear to work reasonably well. Comparison (with a proprietary manufacturer's product) calculation is provided at the bottom of calculator's technical help page, the relevant results from which are thus:
Power: manufacturer: 0.16 kW; Fans (forward-facing): 0.173 kW; Fans (backward-facing): 0.1597 kW
Efficiency (overall): manufacturer: 32.6% Fans (forward-facing): 29.8% Fans (backward-facing): 33.7%
Efficiency grade N: manufacturer: 44% Fans (forward-facing): 32.41% Fans (backward-facing): 43%
Given the cursory (less than 5-minutes) nature of our comparison calculation, together with the lack of manufacturer’s design data (backward-facing blades are most common and therefore assumed to apply here), and the fact that the manufacturer's results apply to a fan (not an impeller), it appears that the calculator is accurate to within acceptable limits - for an impeller (not a fan). However, it would also appear that the manufacturer concerned has achieved a high level of efficiency in its casing design and development.

Whilst CalQlata is not prepared to divulge personal copyright data, we can offer the following advice with his Excel calculations:

As we all know, equipment efficiency is defined as a ratio of output to input of a given parameter. The efficiency of any process will therefore vary according to the parameter(s) selected. The efficiencies calculated in Fans are as defined by Charles Innes. Your efficiency calculation is unlikely, therefore, to match that of Fans unless it is based upon the same parameter(s) and using the same theory. Both efficiencies may therefore be different and yet correct according to their relative parameter(s).

Whilst CalQlata cannot comment on the accuracy or appropriateness of third-party software, perhaps your results apply only to the movement of air, and may not include the requirements for actually achieving this movement, which will vary with impeller design and air velocities. For example;
1) Torque in Fans only applies to a theoretical impeller. If the casing and operational equipment used for the fan in which the impeller is installed are not carefully designed/specified, the fan’s operational torque will be significantly higher than that of the impeller, and the reason CalQlata recommends that its users approach their preferred manufacturer for procurement purposes.
2) The flow of air alone is just one of many factors that affect operational torque in an impeller. Others are; the shape of its blades, impeller aspect ratio, impeller material properties, the inlet and outlet air velocities, skin-friction, inlet areas (diametral vs circumferential), blade size (width vs depth) and shape etc. If these are not taken into account, theoretical torque may vary considerably with that expected.
Hint: drag and added mass play a significant part in the energy required to generate air-flow through an impeller.
3) The heads as calculated in Fans are defined as follows:
H; impeller generated before losses
Hᵀ; impeller generated after losses
Hc; the effect of the casing outlet (based upon relative areas - impeller vs casing).
H & Hᵀ are as defined by Innes and Hc is our own estimate based upon restricted (if this applied to the inlet or outlet orifices) airflow.
The actual head generated by the [impeller] torque at the relevant performance will vary with performance efficiency. If the efficiency is low, the calculated head will also be low and vice-versa. Therefore, it is important to select the most appropriate efficiency parameters when estimating the head generated by the impeller.

It is possible that you may be comparing a practical calculation method that considers the means needed to achieve actual performance with a theoretical calculation method that only addresses the movement of air. If this is the case, it is not surprising that your results are incompatible.

The answer to your question is yes. But a fan comprises two fundamental components; the impeller and the casing.
As explained in the technical help page for this calculator, Fans calculates the performance of impeller, the component that is mathematically predictable.
The casing is not so easily predictable as its effect on fan performance is due to build quality.
Therefore, you use the Fans calculator to determine the performance you need and then build and test the most appropriate design for your casing.

To answer your questions;

Have you read the technical help pages for the calculator: https://www.calqlata.com/productpages/00060-help.html & https://www.calqlata.com/productpages/00060-QandA.html

You must understand that this calculator is based upon the theory of Charles Innes - the original theorist for impeller design. All designs and theories today are based upon his work. But they apply only to the impeller, not the fan – as explained in the above technical help page. The bottom of that page provides a calculation result from this calculator, which compares favourably with a proprietary design, as it should.

This is the least understood by our customers of all theories, and usually takes a while to get used to the input-output data relationships. But you will get to understand it as you play.

Due to the difficulties our customers have understanding Innes’ theories, this has become our most verified and modified (input vs output) calculator, but it is exactly as Innes designed it. It works.

There are a number of reasons why any efficiency can be greater than 100%:
1) The input data is outside the bounds of the theory – so the answer is incorrect
2) The expected efficiency as defined by Charles Innes has been exceeded – the answer is correct
3) One of the input values is incorrect with respect to another input value – you need to play
4) Etc.

Please note: We only provide the calculator, which applies only to the impeller and is correct according the most recognised associated theory. It is up to its users to get the design configuration needed. That is the purpose of all of our calculators, you play with the input values until you get the output value you are happy with. I would, however, advise you to read the above links carefully, they are very helpful.

First I must point out that vi is the overall (theoretical) velocity generated in the air as it passes the inlet tip of the blade.
And vo is the overall (theoretical) velocity generated the in the air as it passes the outlet tip of the blade.
These can be seen in the calculator's diagram.
They are the [theoretical] air velocities across the impeller (only), according to Charles Innes' theory, which remains valid today. The drive power, which is not included in Charles' Innes theory is based upon that necessary to push a blade through air according to Bernoulli. Both of which are just theories, but they are well known and respected.
However, Charles Innes' theory has problems with 90° straight blades. The calculation results are not always reliable. This is reflected in the fact that 90° straight-bladed impellers induce pressure from only centrifugal flow. Not a particularly efficient design.

When efficiencies and casing design are included in the calculation, actual [fan assembly] output data is usually very different.

Fans' calculations are correct and accurate (according to the theory) for the impeller. The casing figures (pc, vc, ρc, Hc, Pc) are only as expected based upon relative [inlet and outlet - impeller and casing] areas.
Your Client is quite correct, however, to use the flow-rate and outlet cross-section to establish overall outlet velocity; vc (87.27) is a theoretical maximum.

Due to his use of 90° straight blades, your Client's design shows a low efficiency (head loss efficiency (%) {air or mechanical efficiency}: εᴴ = -236.095011), which will significantly affect performance.
If you divide vc by the two primary efficiencies achieved (-236.1 & -219.33) the outlet velocity will be about 16.8 m/s. If you then apply casing design efficiencies the actual velocity will no doubt drop by another 1/3rd (11 m/s). As I do not know your Client's casing design efficiency, this is just a guess.

The purpose of Fans was originally to provide the fan designer with a calculator for the impeller only. It is the only part of the fan design that can be accurately predicted with good reliability.
Normally a fan designer will play with the impeller calculations to achieve maximum; efficiency, head, pressure, flow, power, etc. as required, and then design his/her casing to minimise losses. The performance of the assembly (impeller and casing) will be that documented for their clients.
It is impossible for me to include the true effects of the casing as I have no way of knowing its peculiarities. The design ramifications are infinite. I included the casing input ‘Ac’ simply because so many of your clients have a problem separating impeller design from fan design. I'm not sure it was a good idea in hindsight, though.

Yes
If you blocked off the exit you would achieve maximum pressure, but it would not be that defined by the Fans Calculator.
The Fans calculator uses Charles Innes’ theory which relies on the flow of air across the inlet tip of the blades.
If you have no flow, the theory doesn’t apply.

Your client appears to be using lb, ft & R Imperial units in his calculation, together with an input value for Ra of 0.07666666

Metric:
Ri = 8.24992342 J/K/mol
RAM = 0.029324 kg/mol {air}
Ra = Ri/RAM = 281.336905617022 J/kg

Convert:
J > ft.lb = 0.73756215
K > R = 1.8
kg > lb = 2.20462262

Imperial:
Ri = 3.3804618085 ft.lb/R/mole {Ri x 0.73756215 ÷ 1.8}
RAM = 0.064648354 lb/mole {RAM x 2.20462262}
Ra = Ri/RAM = 52.28999061 ft.lb/lb

I would normally expect therefore, that his input value for Ra should be 52.28999061 for air

The problem is the requirement for a high outlet pressure relative to the desired flow rate.

A high (reversed) inlet angle (θᵢ) will artificially increase pressure for best results.

You can play with the input and output diameters and the impeller width to achieve the desired flow rate.

You should remember, however, that the Fans calculator only provides the performance characteristics of the impeller.
Alternatively, You could design an impeller that gives a lower pressure and higher flow-rate and then increase pressure and decrease flow rate by playing with the casing outlet dimensions (PV=RT).

We should point out that a casing outlet with cross-sectional area no smaller than the impeller outlet area will result in the lowest noise-level.

Our Fans calculator calculates the airflow and power consumption for an impeller.
You would have to design the casing/cowling yourself and calculate the effects that would have on the impeller performance.
The casing/cowling options are infinite and do not lend themselves to general calculations.

Fan technology is well established and proven to work for all blade angles that comply with Charles Innes' theory, which has been the industry standard since 1916
Impeller design is difficult to understand if you don't take the time to try and understand the basic principles.
Performance is as much dependent upon casing design as it is on the impeller. The Fan calculator only calculates the performance of the impeller.
It is difficult to accommodate casing design as the options are infinite. However, the principle design requirement for a casing is the relative areas between impeller outside perimeter and casing outlet. A larger casing outlet will assist flow, a smaller outlet will assist pressure.

The customer concerned set his blade outlet angle such that it is driving air back into the impeller with greater energy than his inlet angles are able to overcome. He has completely misunderstood the basic principles of driving air through an impeller and refuses to accept the fact.

Inlet blade angles greater than 90° will not drive air out through an impeller, they will drive it back into the inlet cavity.

Moreover, if you do not set your inlet blade angles shallow enough to provide sufficient positive outward drive to overcome inward drive from outlet blade angles greater than 90° the theory will become unstable, as would be such an impeller.

It is important you try to understand the behaviour of air as it passes through the fan. Whilst it is largely based on common-sense, if you ignore basic flow characetristics you will never get your impeller to work, theoretically or practically.

Please take look at the tips we provide in our technical help page

As a result of this question, CalQlata successfully carried out an internal verification based upon the energy required to shift such a mass.

Whilst it was considered prudent to re-issue the 'Fans' calculator now calculating the impeller speed (RPM) required to generate an entered value for volumetric flow rate based upon the required energy;
the aforementioned rule of thumb still applies.

The standard theory on centrifugal fans was generated by Charles H Innes in 1916 ("The Fan"), and it appears to have stood the test of time.
His theory is based upon the aerodynamics of the blade transferring air from the toe of the blade to the outer lip in a single revolution of the impeller.
He also states that the greater the number of blades the more uniform the flow; i.e. more of the airflow will be laminar (less turbulent). An infinite number of blades will generate the most uniform flow. However, he also states that this comes with increased skin friction.
At some point the losses from skin friction will exceed the inlet and outlet losses and a compromise is needed.
Innes suggests that six blades is the best for a low aspect ratio (<0.67) increasing to twelve blades for aspect ratios approaching one.
This recommendation is based upon his own (extensive) experience.

Twelve blades, however, may not be suitable for very large diameter impellers of high aspect ratios.
In my own (less extensive) experience, I have settled for five blades for aspect ratios of 0.5 increasing to an inlet pitch of similar dimension to the radial depth of the blade.
I would multiply the skin friction in the calculation by the number of blades and would also use a higher value than that suggested by Innes for frictional resistance (0.125) on the basis that 'used' fan blades (that have been in service for some time) will have slightly eroded surfaces (i.e. 0.15 to 0.2).
With regard the expected accuracy of the calculator, if Innes is correct, then the calculations in CalQlata's Fans calculator should be ±0%. But I would only consider this to be the case for properly designed impellers in dry air. Moisture in the air (>1%) can cause a reduction in efficiency. In normal everyday conditions, with a used fan that has been properly designed and maintained, I would expect an accuracy better than ±10% (i.e. ±4% to ±8%), but this does not include the effects of the inlet and outlet diffusers, which can improve or reduce the impeller's effective efficiency.

I (personally) have never seen a fan deliver more than its impeller volume per revolution unless the atmospheric outlet pressure is less than the inlet pressure (an effective vacuum). Therefore, I am unable to refute Innes' theory. I am happy, however, to leave this debate open to anybody that can show Innes' theory to be incorrect. Please let us know if you can do this mathematically and supported with practical evidence.

The theories in the calculator are correct for all sizes, i.e. for fans with impellers smaller than a millimetre to greater than 10m.
The problem with ever decreasing size is friction.

As your fan gets smaller the ratio of surface area with volume increases, and the smaller it gets the greater this ratio becomes.
If we use a pipe as an analogy:
Dia   Area               Volume           A:V        δε
24   150.7964474   1809.557368   0.083
23   144.5132621   1661.902514   0.087   4.167%
22   138.2300768   1520.530844   0.091   4.348%
21   131.9468915   1385.442360   0.095   4.545%
20   125.6637061   1256.637061   0.100   4.762%
19   119.3805208   1134.114948   0.105   5.000%
18   113.0973355   1017.876020   0.110   5.263%
17   106.8141502   907.9202769   0.120   5.556%
16   100.5309649   804.2477193   0.125   5.882%
15   94.24777961   706.8583471   0.13 0  6.250%
14   87.9645943   615.75216010   0.143   6.667%
13   81.68140899   530.9291585   0.154   7.143%
12   75.39822369   452.3893421   0.167   7.692%
11   69.11503838   380.1327111   0.180   8.333%
10   62.83185307   314.1592654   0.200   9.091%
 9   56.54866776   254.4690049   0.220   10.000%
 8   50.26548246   201.0619298   0.250   11.111%
 7   43.98229715   153.9380400   0.286   12.500%
 6   37.69911184   113.0973355   0.330   14.286%
 5   31.41592654   78.53981634   0.400   16.667%
 4   25.13274123   50.26548246   0.500   20.000%
 3   18.84955592   28.27433388   0.670   25.000%
 2   12.56637061   12.56637061   1.000   33.333%
 1   6.283185307   3.141592654   2.000   50.000%

In this case ‘δε' represents the increase in inefficiency over the previous size
As A:V increases the ratio of surface area to volume increases, and this increase is exponential. As you can see from the table when the pipe size falls to 3" the increase becomes rapid. Look at the difference between a 24" to 23" (4%) and that for 4" to 3" (25%)

Surface friction has a far greater effect on efficiency in a fan than it does in a pipe because the ratio of surface area (contact surface) is greater than in a pipe. Therefore, a similar table to that above for fans would show an even more marked increase at smaller diameters.

Centrifugal fans are less suited to smaller diameter for a number of reasons.
1) The surface area in contact with the flowing air is greater than for axial fans
2) The air is forced to change direction
3) The shape of the inlet and outlet ducts is not best suited for reducing frictional losses (circular is better than rectangular)

This is why centrifugal fans tend to be targeted for larger fans and axial configurations for smaller diameters
I am not saying you shouldn't try to design a small centrifugal fan if that best suits your purposes, simply that you take extreme care in its design. For example, the following should be considered:
1) Use a material with a very low surface friction (or coat the material, but bear in mind that the loss of the coating during the design life will result in a loss of efficiency).
2) Reduce the number of blades
3) Minimise the surface area of material in contact with the flowing air
4) Eliminate sudden changes in shape

For what it’s worth, if I were designing a very small high-performance fan, I would start with a multistage axial configuration (along with suitable venturies if I was looking for pressure as opposed to flow).

Follow-Up Email:

I believe that your input/output data is based upon the following units:
Input Data:
θᵢ,   45 {°}
θₒ,   120 {°}
Øᵢ,   0.02 {m}
Øₒ,   0.075 {m}
w,   0.01 {m}
ρᵢ,   1.165 {kg/m³}
pᵢ,   101322.5 {N/m²}
Ṯ,   291 {K}
g,   9.80663139 {m/s²}
Rₐ,   8.3143 {J/K/kg}
F,   0.125
N,   7000 {RPM}
Output Data:
Q,   0.004788 {m³/s} [287.28 l/min]
H,   38.207032 {m}
δp,   436.50484 {N/m²}
pₒ,   101,759.00 {N/m²}
ρₒ,   42.058538 {kg/m³}
ε,   47.55515 {%}
T,   0.002851 {N.m}
P,   4.394551 {N.m/s}
A,   0.001486 {m²}
v,   3.221569 {m/s}
Lˢ,   0.00427 {m}
Lᶠ,   0.035085 {m}
Lᵉ,   42.096188 {m}
vᵢ,   7.625286 {m/s}
vₒ,   28.734015 {m/s}
v₁ᵢ,   7.619792 {m/s}
v₁ₒ,   2.031945 {m/s}
v₂ᵢ,   7.330382 {m/s}
v₂ₒ,   27.488935 {m/s}
v₃ᵢ,   10.776012 {m/s}
v₃ₒ,   2.346287 {m/s}
v₄ᵢ,   -0.289409 {m/s}
v₄ₒ,   28.662079 {m/s}

With regard to your concerns about flow rate:
Q = 0.004788m/s = 287.28 l/min
A very good rule of thumb for any fan is that its impeller will pass its (contained) volume of air in one revolution.
If I apply this argument to the volume of your impeller, I get: 287.2593783 l/min
So the calculator and the rule of thumb both appear to be working correctly.
However, the above would only be achievable if chamber, impeller, inlet and outlet designs added nothing to the losses#. As mentioned before, their influence becomes significantly greater as the size of the fan reduces. Like you, I would expect closer to 100 l/min for a centrifugal pump of the size you are designing unless all aspects of the fan design were perfect in every way. In fact, even with the best possible materials and designs, I would not expect see better than 200 l/min for such a small (centrifugal) fan
With larger fans (i.e. 6" and greater), it should be much easier to achieve the calculated values.

May I venture a couple of comments on the input data?
The value of Rₐ is for the specific gas constant;
Rₐ (Rᵢ/RAM) = 283.5312934 J/K/kg
as opposed to ideal gas constant;
Rᵢ = 8.3143 J/K/mole
You can see how we use these symbols in our definitions page {https://www.calqlata.com/help-definitions.html > Gas Constant}
The differences you will notice are in outlet density {ρₒ}, the minimum area diffuser requirement {A} and all the velocities {v}

The recommended angle for the blade inlet should be used where possible as you will see improvements in efficiency, outlet pressure, outlet velocity and power consumption. Whilst I agree that such improvements are very small between 45° to 46.109°, every little helps when attempting to minimise defects (for such small fans).

I notice that the outlet angle (120°) has turned the blades from backward facing to forward facing (see https://www.calqlata.com/productpages/00060-help.html Fig 3). I am not sure if this was intentional (special requirements) but the smaller the outlet angle for a backward facing blade, the better its efficiency.
Using your customer’s input data with θᵢ = 46.109° and θₒ = 8° the efficiency increases from 47.5% to 72% and the power required to run the fan drops from 4.4W to 2W. Whilst you lose head and pressure the efficiency gain is greater than the loss.
By setting the outlet angle (θₒ) to 25° you achieve virtually the same head and pressure but with an efficiency of over 57% and a drop in power consumption of 25%

# Note: the efficiency quoted (ε {%}) is for blade design only