What is the difference between a conceptual design and a conceptual model?
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As others asserted the terms can be ambiguous, depending on context.
From my experience:
A non-committing design, that often captures high-level ideas without the details.
"For this specific problem with have sketched 5 conceptual designs."
The name also stands for a phase during the creativity/problem solving process - once the problem is understood we move to the idea generation phase and diverge by coming up with many alternative conceptual designs. After that we start converging by analysis and synthesis and the optimal solution than goes through a detailed design.
The field of conceptual modelling is vast. It is a multi-disciplinary field dealing with modelling of concepts.
"To understand technical debt better, we devised this conceptual model."
The guru in (and to some extent the godfather of) this field, John Mylopoulos, categorises conceptual models into:
All these constitute a conceptual model:
In fact, you can argue that nearly every diagram that involves nodes that represent concepts and their relationships is a conceptual model.
Here's a conceptual model capturing the social aspects of agile development:
In a sense, they could be considered two phrases for the same thing. The core difference is that a conceptual model is more nebulous than a conceptual design. Conceptual design implies visualization, which while it can still be very basic (doodles on a cocktail napkin), is still somewhat more "realized" than a conceptual model. A conceptual model can be virtually anything.
For example, a conceptual model for a user signing into a website could be:
User enters username and password and clicks a button to submit
Whereas a conceptual design might be a crude drawing of the sign in form.
I think the question itself is rather ambiguous. Looking at the definition of conceptual (Google define search):
relating to or based on mental concepts.
and then the definition of concept (Google define search):
an abstract idea
If you use the term conceptual with model in its definition (Google define search):
a thing used as an example to follow or imitate
Then you get "a thing used as an example to follow or imitate something relating to or based on mental concepts".
And the term conceptual with design in its definition (Google define search):
a plan or drawing produced to show the look and function or workings of a building, garment, or other object before it is made
Then you get "a plan or drawing produced to show the look and function or workings of something relating to or based on mental concepts".
I think what this all means is that a conceptual design could describe a conceptual model, and a conceptual model and be translated into a conceptual design. But in either case you still need some way to realize a conceptual model or a conceptual design (i.e. implementation).